Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
METHOD FOR MEASUREMENT OF RESIDUAL STRESS AND COEFFICIENT OF
THERMAL EXPANSION OF LAMINATED COMPOSITES
By
DONALD G. MYERS II
A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2004
Copyright 2004
by
Donald G. Myers II
This thesis is dedicated to my family, loved ones, and in the memory of my father.
ACKNOWLEDGMENTS
I would like to thank my advisor, Dr. Peter Ifju, for all of his support and advice. I
would also like to thank my other committee members, Dr. Raphael Haftka and Dr.
Bhavani Sankar, for their advice. Thomas Singer, William Schulz, and Lucian Speriatu
were fellow assistants in the Experimental Stress Analysis Lab that provided invaluable
assistance in my efforts. I would also like to extend a thank you to Dr. Leishan Chen for
all of his assistance. Together all of these people along with my various professors have
made my time at the University of Florida a wonderful educational experience. Finally I
would like to say thank you to my entire family and loved ones for everything they have
done for me throughout my life.
iv
TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................................................................................. iv
LIST OF TABLES........................................................................................................... viii
LIST OF FIGURES ........................................................................................................... ix
ABSTRACT...................................................................................................................... xii
CHAPTER
1 INTRODUCTION ........................................................................................................1
1.1 Historical Background ............................................................................................1 1.2 Failure of the X-33 Fuel Tank ................................................................................2 1.3 Residual Strains and Stresses..................................................................................6 1.4 Structural Reliability...............................................................................................7 1.5 Research Objectives................................................................................................9
2 RESIDUAL STRESS CHARACTERIZATION TECHNIQUES..............................10
2.1 Introduction...........................................................................................................10 2.2 Destructive Testing Techniques ...........................................................................11
2.2.1 Hole-Drilling Method.................................................................................11 2.2.2 Cutting Method...........................................................................................12 2.2.3 Ply Sectioning Method ...............................................................................12 2.2.4 First Ply Failure ..........................................................................................13
2.3 Non-Destructive Techniques ................................................................................13 2.3.1 Warpage......................................................................................................13 2.3.2 Embedded Strain Gages .............................................................................14 2.3.3 X-ray Diffraction ........................................................................................15 2.3.4 Electrical Resistance...................................................................................15 2.3.5 Cure Referencing Method ..........................................................................15
3 CURE REFERENCING METHOD...........................................................................17
3.1 Selected Laminates ...............................................................................................17
v
3.2 Cure Referencing Method.....................................................................................18 3.3 Moiré Interferometry ............................................................................................19 3.4 Diffraction Gratings..............................................................................................20 3.5 Curing Procedure ..................................................................................................22 3.6 Moiré Measurement Technique............................................................................27 3.7 Data Analysis........................................................................................................29 3.8 Error Sources ........................................................................................................30 3.9 Conclusions...........................................................................................................32
4 STRAIN GAGE METHODS .....................................................................................34
4.1 Acquisition Theory ...............................................................................................34 4.2 Specimen Preparation ...........................................................................................35 4.3 Testing Procedure .................................................................................................38
4.3.1 Cryogenic Tests ..........................................................................................39 4.3.2 Elevated Temperature.................................................................................42
4.4 Data Analysis and Results ....................................................................................44 4.5 Error Sources ........................................................................................................47 4.6 Conclusions...........................................................................................................48
5 MATERIAL PROPERTY TESTING.........................................................................51
5.1 E1 Testing..............................................................................................................51 5.2 E2 Testing..............................................................................................................52 5.3 G12 Testing............................................................................................................54 5.4 Data Analysis and Results ....................................................................................55 5.5 Micro-mechanical Methods ..................................................................................58 5.6 Error Sources ........................................................................................................59 5.7 Conclusions...........................................................................................................59
6 PLY LEVEL RESIDUAL STRAINS AND STRESSES ...........................................61
6.1 Residual Strain Calculation ..................................................................................61 6.2 Residual Stress......................................................................................................62 6.3 Conclusions...........................................................................................................64
7 CLASSICAL LAMINATION THEORY...................................................................65
7.1 Theory Development ............................................................................................65 7.2 Experimental vs. Analytical Results.....................................................................72 7.3 Conclusions...........................................................................................................76
vi
APPENDIX A CRM IMAGES OF SPECIMENS..............................................................................77
B MATLAB CLT CODE...............................................................................................85
LIST OF REFERENCES...................................................................................................98
BIOGRAPHICAL SKETCH ...........................................................................................103
vii
LIST OF TABLES
Table page 3-1 Total surface strain on unidirectional laminates at 24°C.......................................30
3-2 Total surface strain on optimized angle ply at 24°C..............................................30
3-3 Total surface strain on RLV laminate at 24°C.......................................................31
4-1 Average COV across temperature range................................................................49
4-2 Unidirectional chemical shrinkage induced strain.................................................50
4-3 OAP chemical shrinkage induced strain ................................................................50
4-4 RLV chemical shrinkage induced strain ................................................................50
5-1 Results of each tension test ....................................................................................58
5-2 Comparison of experimental measurements and micro-mechanics values ...........59
viii
LIST OF FIGURES
Figure page 3-1 Schematic of a four-beam moiré interferometer....................................................20
3-2 Image of moiré interferometer used in testing.......................................................20
3-3 Replication schematic ............................................................................................23
3-4 Tool separation device ...........................................................................................24
3-5 Teflon coated epoxy film casting device ...............................................................24
3-6 Autoclave oven ......................................................................................................26
3-7 Vacuum bag assembly schematic ..........................................................................26
3-8 IM7/977-2 cure cycle.............................................................................................28
3-9 Moiré grating on specimen ....................................................................................28
4-1 Schematic of strain gage alignment .......................................................................37
4-2 Gauging of sectioned specimen .............................................................................37
4-3 Gauging of reference materials..............................................................................38
4-4 Environmental chamber and LN2...........................................................................40
4-5 Laminate coupons wired and placed inside environmental chamber ....................40
4-6 Quarter-bridge schematic.......................................................................................41
4-7 Unidirectional laminate surface strain ...................................................................45
4-8 OAP laminate surface strain ..................................................................................46
4-9 RLV laminate surface strain ..................................................................................46
4-10 Average of laminate surface strains.......................................................................49
4-11 Coefficient of thermal expansion...........................................................................50
ix
5-1 E1 and E2 testing fixture.........................................................................................53
5-2 Shear test specimen schematic...............................................................................56
5-3 Shear Testing Fixture.............................................................................................56
5-4 E1 stress/strain curves ............................................................................................57
5-5 E2 stress/strain curves ............................................................................................57
5-6 G12 stress/strain curves...........................................................................................58
6-1 Residual strain on selected plies ............................................................................63
6-2 Residual stress level on selected plies ...................................................................63
7-1 Laminate coordinate system ..................................................................................67
7-2 Layer schematic and nomenclature........................................................................67
7-3 Unidirectional CLT and experimental results........................................................73
7-4 OAP CLT and experimental results.......................................................................73
7-5 RLV CLT and experimental results.......................................................................74
7-6 Unidirectional CLT shifted for chemical shrinkage ..............................................74
7-7 OAP CLT shifted for chemical shrinkage .............................................................75
7-8 RLV CLT shifted for chemical shrinkage .............................................................75
A-1 Unidirectional Laminate Run 1..............................................................................77
A-2 Unidirectional Laminate Run 2..............................................................................78
A-3 Unidirectional Laminate Run 3..............................................................................78
A-4 Unidirectional Laminate Run 4..............................................................................79
A-5 Unidirectional Laminate Run 5..............................................................................79
A-6 OAP Laminate Run 1.............................................................................................80
A-7 OAP Laminate Run 2.............................................................................................80
A-8 OAP Laminate Run 3.............................................................................................81
A-9 OAP Laminate Run 4.............................................................................................81
x
A-10 OAP Laminate Run 5.............................................................................................82
A-11 RLV laminate Run 1 ..............................................................................................82
A-12 RLV Laminate Run 2.............................................................................................83
A-13 RLV Laminate Run 3.............................................................................................83
A-14 RLV Laminate Run 4.............................................................................................84
xi
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science
METHOD FOR MEASUREMENT OF RESIDUAL STRESS AND COEFFICIENT OF THERMAL EXPANSION OF LAMINATED COMPOSITES
By
Donald G. Myers II
May 2004
Chair: Peter G. Ifju Major Department: Mechanical and Aerospace Engineering
The development of residual stresses in laminated composites is a very important
area of extensive research. Residual stress can dramatically reduce the strength of a
component built of a laminated composite. The existence of these residuals stresses and
strength degradation can lead to catastrophic material failures as seen in the X-33. A
method has been proposed to measure and quantify these stresses as well as determine the
thermal expansion coefficient of laminated composites as a function of temperature. This
method relies on the combination of existing techniques in a manner that will allow for
the acquisition of stress values across a wide range of temperatures. The first technique
employed is a novel strain measurement method know as the Cure Referencing Method.
This technique is based on transferring a moiré grating onto a specimen during the cure
process. The grating becomes a part of the specimen at cure temperature during the
polymerization of the resin and therefore carries all strain information from the curing
process including temperature dependent strains as well as chemical shrinkage
xii
information. The tool with which the grating was transferred to the specimen is used as a
reference from which all relative deformations are calculated. This process is
implemented on laminates of three differing stacking sequences, each of which provides
its own unique data point. The results from this method provide data points from which
the next technique references the strains and stresses as a function of temperature. After
strain measurements are taken from the moiré specimen, they are fitted with strain gages
in the x and y directions. Strain values are observed as a function of temperature as the
specimens are exposed to temperatures between cure and cryogenic conditions. The
strain gage readings are compensated for temperature effects and are referenced to the
strain values at room temperature given by the cure referencing method. The resulting
values of strain represent the total surface strain at any given temperature for the various
laminates. The surface strain on the unidirectional laminate represents the free expansion
state any lamina would reach if not constrained. This value is used to determine residual
strain present in different layers of the multidirectional laminates. The coefficient of
thermal expansion is also calculated as a function of temperature from the strain gage
readings. Basic material testing is also conducted and the values of longitudinal,
transverse, and shear moduli are presented. Variability in these measurements is also
reported in conjunction with other work at the University of Florida to perform reliability
based optimization. Once the residual strain and material properties are calculated they
are transformed into stress by the usual stress-strain relations. The final ply level residual
stress is plotted as a function of temperature. The results of surface strain measurements
are compared to classical lamination theory analytical predictions.
xiii
CHAPTER 1 INTRODUCTION
Throughout history one of the primary goals of engineers has been to develop
structures and materials that are lighter, stronger, and tougher than those that already
exist. The introduction of advanced structural composites (ASCs) has helped to achieve
that goal. Composite materials are those that consist of two or more separate materials
that are combined macroscopically into a structural unit [13]. The prevalence of these
composite materials has skyrocketed since the 1960s and are now used in structures as
advanced as satellites, space structures, and military aircraft and in everyday sporting
goods around the house such as skis and golf clubs.
1.1 Historical Background
Despite the boom in significance since the development of ASCs in the 1960s,
basic man-made composite materials have been around for thousands of years. Some
examples can be seen as early as the times of the great Egyptian kingdoms when brick
was made of clay and straw, and in South and Central America where the people used
plant fibers in native pottery [13]. Both of these are examples of one of the most
predominant composite materials, fiber reinforced composites. Other examples of fiber-
reinforced composites include glass/epoxy, steel-reinforced concrete, and graphite/epoxy,
the subject of this investigation.
The motivation for the use of fiber-reinforcement in materials comes from research
by Griffith published in 1920 that demonstrated that some materials are stronger and
1
2
stiffer as fibers than in bulk form [14]. Some reasons for this increase in strength include
less material flaws, orientation of molecular chains, and lower dislocation densities.
Along with all the obvious advantages, advanced fiber-reinforced composite
materials have their shortcomings as well. A primary performance weakness is that most
of the strength exists in the direction of the fiber and the material performs poorly in
transverse directions. Transverse reinforcement is necessary to make the material useful
and this is done usually by orienting the fibers at various angles.
Using composite materials also poses other problems for designers. These
materials are much more difficult to analyze and characterize. Many of the design and
analysis techniques that have been used for years on other materials cannot be applied
directly to composite materials. Because they consist of more than one material they are
often orthotropic in nature. Also, once several layers of lamina are assembled into a
laminate, to correct the transverse weakness or tailored to achieve desired directional
strengths, a complex interaction between layers develops. This is a macroscopic issue
while on the microscopic level there is also a complex interaction that takes place within
each lamina between the fiber and matrix material.
Due to these analysis difficulties, the behavior of composite materials is not fully
understood and this lack of understanding can lead to devastating mechanical failures. It
is the goal of this paper to detail a combination of methods that can be used to
characterize a material system and then provide a detailed set of data and information on
the IM7/977-2 material system.
1.2 Failure of the X-33 Fuel Tank
For several years, efforts have been underway to develop a reusable launch vehicle
(RLV) to replace the aging space shuttle fleet. The goal is to develop a vehicle that is
3
entirely self-contained without the need of external fuel and propulsion components such
as the external fuel tank and solid rocket boosters seen on the current space shuttles. The
vehicle is to be able to takeoff, perform its mission in space, and return to earth entirely
under its own power. The development of the X-33 was an attempt to achieve these
goals.
The design of the X-33 called for an internal liquid hydrogen (LH2) fuel tank [12].
This tank was to be an integrated part of the structure designed to provide both fuel
storage and carry structural loads. One of the most important factors in the design of an
RLV is to reduce the takeoff weight as much as possible. Manufacturing a tank
consisting of ASCs was seen as the best way to do this.
The X-33’s fuel tank and aft fuselage were designed to consist of two LH2 fuel
tanks, with each composed of several “lobes” made of graphite/epoxy and having bonded
woven composite joints. Each lobe was a sandwich panel with face sheets consisting of
the IM7/977-2 material system and a Korex® core. The tanks were to store the LH2 fuel
as well as be part of the primary aft structure designed to distribute thrust, inertial,
landing gear, and aero-control surface loads.
The first of two tanks was designed and delivered by Lockheed Martin to NASA at
the Marshall Space Flight Center for preflight testing. The preflight testing was designed
to verify that the tank was able to withstand the cryogenic loading conditions of –423º F
(20º K) as well as the expected flight loads. The planned testing sequence consisted of:
1. A partial liquid nitrogen (LN2) fill and proof test to 32 psig, followed by a leak check at 34 psi of gaseous helium
2. A LH2 fill and structural load application. The test was to consist of six conditions: a proof pressure test at 42 psig and five loading tests at 5 to 42 psig
3. A final 34 psig gaseous helium test
4
It would be seen soon after testing began that the design was not feasible for its intended
purpose.
In early November of 1999 the fuel tank began the first combined mechanical
loading and LH2 storage test. It had been previously been filled and leak tested with LN2
and LH2 in the absence of external loads. This time the initial pressure and loading tests
were successful and the tank was drained of LH2. Fifteen minutes after the tank was
drained a major mechanical failure took place as the outer facesheet and core material
debonded from the inner facesheet. The following lists the observations that were made
during the test and time leading up to the failure.
The first observation was that at the time of fueling there was an expected decrease
in the pressure of the sandwich core resulting from the temperature change within the
tank. The second observation was an unanticipated increase in the core pressures as the
tank neared full capacity and pressurization. A third observation was the continued
increase in core pressures as the tank was drained. This event was expected, as the
temperature increase would reverse the initial trend seen in observation one and pressure
should have returned to ambient conditions. Fourth was the incident of the composite
sandwich failure. At the time of the failure core pressures were between 50 and 60 psia.
These pressures rapidly dropped to atmospheric as the region failed. Finally, the
remaining sections of the core that did not experience a failure had internal pressures
above the expected ambient a full 13 hours later.
Review and analysis of the data resulted in the following failure sequence. The rise
in pressure in the second observation was most likely the result of seepage of H2 into the
core through the inner facesheet. The method of transportation was though mirocracks
5
that had developed in the inner facesheet due to the stresses and strains cause by the
cryogenic environment. As the tank was filled with LH2, the cooling caused a vacuum in
the core, which helped to pull in GH2 until the area was covered with the liquid. LH2 was
then also able to penetrate the facesheet through the microcracks. As the tank was
emptied some of the fluid was able to travel back through the microcracks and be
removed from the tank. However, before all of the fluid had passed back into the interior
of the tank the pressure and strain decreased closing the microcracks and trapping some
of the fluid inside the core. As the temperature continued to climb the pressure of the
vaporizing H2 resulted in a buildup until the outer facesheet and core debonded from the
inner facesheet.
The properties of the laminates were estimated from existing databases. Also it
was assumed that 5000 microstrain (µε) was an adequate allowable level to limit
permeability. The above failure observed in the X-33 project is a perfect example of how
the behavior of ASCs are not fully understood and how this lack of information can result
in catastrophic failures. The design group failed to adequately cover combined
thermal/mechanical effects. It is this thermal effect, or processed induced residual
effects, that are not fully understood and result in considerable structural issues.
The failure of the X-33 and what it means to the future of composite
implementation is the driving force behind this thesis. This work will propose combined
methods that can be used to better evaluate the IM7/977-2 of the X-33 and other
composite material properties as they are subjected to extreme thermal conditions. This
material characterization can be used to remove the some of the assumptions that are
6
made in current composite design and replace them with concrete experimentally
determined values.
1.3 Residual Strains and Stresses
Root failure of the X-33 was most likely the failure to understand and accurately
predict the residual strains and stresses that develop in a composite as a result of curing
and the exposure to changing temperatures, especially low temperatures. When a
polymeric fiber reinforced composite is manufactured it usually undergoes a process
where the resin is heated, the fibers are wetted, and cure takes place at high temperatures
[37]. During this process and subsequent conditions residual strains and stresses will
develop within the laminate.
The developing stresses arise on two different scales. Stresses will occur that
develop between the fiber and matrix in each lamina of a composite. The study of these
effects is termed micro-mechanics, where a laminate is examined on the fiber scale.
Another area of interest is the examination of what happens to a composite on the macro-
mechanical or ply scale. It is on this scale that the following research will focus.
On the ply scale residual strains and stress develop due to the mismatch in the
thermal expansion coefficients of the fiber and matrix as well as the development of
chemical shrinkage. As a lamina is cured its matrix constituent undergoes
polymerization. Epoxy resins undergo condensation polymerization, where two reacting
monomers are brought together to form a new molecule of the desired compound [13]. In
the case of many ASCs this is a two-step process. The production of prepreg tape
consists of wetting the fibers and allowing the matrix to partially cure. The
polymerization process concludes when these prepreg materials are arranged into the
desired stacking sequences and then are heated to the desired cure temperature. As this
7
process takes place the matrix undergoes a volumetric change known as chemical
shrinkage while the fibers remain volumetrically unchanged. This event adds to the
mismatch in expansion of the fibers and matrix, where the matrix is subject to higher
expansion and contraction than the fiber.
Residual stresses develop when the desired expansion of the lamina is restricted.
As the angles of the lamina are varied from ply to ply, the lower contraction of the fiber
will constrain the contraction of the matrix. When temperatures are decreased, as in the
case of the X-33, the matrix desires to contract but instead is subjected to tensile stress as
this deformation is opposed. If all of the fibers are aligned with each other there is no
stress present on the ply scale. A cross-ply [0/90] stacking sequence will result in the
highest level of residual stress.
The residual stresses that develop during processing and operating conditions are
not negligible. They can consume large amounts of a laminate’s strength. If these
stresses are not accurately understood they can lead to drastic material failures as in the
case of the X-33, where the tensile stresses that developed in the matrix material likely
exceeded its critical tensile strength. Once this occurred, micro-cracks were able to
develop allowing the seepage the hydrogen into the core. The development of
microcracks in other structures will allow the fibers to be exposed to possibly degrading
environmental conditions as well as possible chemical attack in storage facilities.
1.4 Structural Reliability
The existence of the overwhelming thermal stresses that resulted in the failure of
the X-33 demonstrates the need to better understand the nature of composite materials
and appropriately design for their implementation. The examination of structural
8
reliability of a component during the design process is a step that must be taken to
prevent such failures.
Previously, the design process worked to build reliability based on a factor-of-
safety approach [48]. This is a deterministic approach to the issue of reliability where
worst case scenarios are proposed, investigated, and accounted for by adding a safety
margin in the design. This approach has been used for years in metallic structures and
can provide adequate reliability for composites but not to the same degree. This is
because composite material performance exhibits a larger statistical distribution and
variation than metals. Reliability-based design methods are being developed to account
for this problem.
Probabilistic design methods are moving to the forefront in composite design.
Probabilistic design is an integrated process that works to define and develop functional
relationships between various design factors and material properties. A statistical
variation in one variable will have consequences to the others and those changes can be
designed for. The probabilistic approach uses statistical characterization instead of
proposed worst case values to define a variable and determine its magnitude and
frequency. The amount of data and how well the variables are defined will influence the
extreme values.
A typical probabilistic design approach has the following fundamental elements:
1. Identify all possible uncertain variables on all scales of the structure, including all variables at the constituent scale, all stages of fabrication and assembly, and the possible applied loads.
2. Assign a probabilistic distribution function for each variable.
3. Process all random variables through and analyzer that can handle micro- and macro-mechanics laminate theories, structural mechanics, and probability theories.
9
4. Extract useful information from the analysis and compare them to probabilistic design parameters.
To be able to carry out the above procedure the following aspects must be characterized as random variables:
• Material mechanical properties • External loads during operation • Manufacturing process variability • Environmental effects • Environmental history during operation • Effect of flaws or damage locations • Predictive Accuracy
The probabilistic approach to analysis and design has become a significant area of
academic interest. Qu et al have investigated areas of interest that are particularly of
importance to the future of RLV design. They performed reliability-based optimization
of composite laminates for cryogenic environments [41]. It was found that under these
conditions that the optimized weight was sensitive to transverse lamina strength and
thermal stresses as they are related to CTE and lay-up sequence. When the confidence in
these values is low the thickness of the laminate is significantly greater leading to a
higher vehicle weight.
1.5 Research Objectives
The work of Qu et al has been carried on at the University of Florida and parallels
the work in this thesis. The goals of this thesis were to establish a simple testing method
to determine residual strains, stresses, and coefficient of thermal expansion as a function
of temperature, and to measure these properties on the IM7/977-2 material system that
was used in the X-33 fuel tank development. In addition to this, various moduli were to
be experimentally determined and statistical data was to be supplied to the optimization
group for further probabilistic analysis and design.
CHAPTER 2 RESIDUAL STRESS CHARACTERIZATION TECHNIQUES
2.1 Introduction
Since the introduction of advanced structural composites the need to understand
their behavior has lead to extensive research in the area of residual stress determination.
Many existing methods have been devised to aid in residual stress characterization.
Some of these methods have been derived from existing tests used on other materials
while others have been developed from scratch. Each of these methods has it own
advantages and disadvantages. As a whole they can be useful in characterizing certain
aspects of residual stress and its effects. Most do have the same substantial flaw that will
be pointed out in this research. None, with the exception of the Cure Referencing
Method (CRM), developed at the University of Florida and used extensively in this work,
have the ability to account for the residual stress that results from the chemical shrinkage
involved in the curing of an ASC. This chapter will give a brief history of and detail the
existing traditional methods that are often used to characterize residual stress in laminated
composites.
Existing methods are usually grouped into two broad groups, those that are
destructive in nature and those that are non-destructive. Destructive testing usually
involves damaging or removing a section of material from a test specimen. The result is
a specimen that can no longer be used for most applications. Non-destructive tests are
generally preferred over destructive tests for obvious reasons. A specimen exposed to
non-destructive means of testing has not lost any usefulness as a structural component.
10
11
Tests can also be repeated on the same specimen to ensure accuracy. These methods also
often provide a means with which to test the effects of time on a component.
2.2 Destructive Testing Techniques
2.2.1 Hole-Drilling Method
One of the oldest and most accepted methods for the determination of residual
stresses is that of the hole-drilling technique. Mathar first proposed this technique as
early as the middle 1930s [33]. Since that time it has been employed to characterize
residual stresses present in metal structures and other isotropic materials. When a hole is
introduced into a stressed body the stresses are relaxed and become zero. This results in
a change in the surrounding strain field, which can be measured and correlated to the
relaxed stresses. This was the heart of Mathar’s work.
The most commonly used application of the hole-drilling techniques involved
drilling a small blind hole near the application point of a strain gage rosette [35]. The
rosette can then measure the relaxed strain and the data can be used to calculate the
residual stress that was present before the creation of the hole. Since the middle to late
1960s efforts have been underway to extend the use of this technique from isotropic to
orthotropic materials [1,28,42,43]. The extension of this method into orthotropic
materials does not yield results as readily as in isotropic materials. The solution method
is numerically intense. Many assumptions must be made in order to simplify the
resulting solutions. The highly orthotropic nature of composites also adds further
difficulty in the measurements themselves. Obtaining measurement precision around the
hole is extremely difficult especially in the fiber direction, even for high precision
techniques such as moiré interferometry [36].
12
2.2.2 Cutting Method
The cutting method is another technique that relies on the same principles as those
in the hole drilling method. Again, stress is relaxed by the removal of material. This
time a notch is removed from a specimen resulting in the creation of a free edge. Lee
devised a method where a moiré interferometry grating was applied to the specimen to
record the resulting strain field [30,31]. The resulting strain field was calculated and
related to the residual stresses by the used of finite element analysis.
This method is not without its faults and has yet to be fully accepted in the
experimental community. Niu poses many questions based on the application of the
grating [37]. The grating was applied after an edge of the composite had been trimmed.
This resulted in some of the residual stresses being released prior to data recording and
the creation of a very complex strain field under the surface. The resulting measurements
that were made contained data that was based on a stress field different from the original
residual stress.
A second technique derived to employ the cutting method was proposed by
Sunderland et al. [46]. The successive grooving technique involved cutting a groove
through the thickness at successive depths. Strain gages were placed opposite the groove
and recorded the changing strain field. The residual stress was then calculated from the
strain by use of a numerical 2-D model for each layer.
2.2.3 Ply Sectioning Method
Yet another destructive technique devised to study residual stresses in laminated
composites has been proposed by Joh et al. [24]. This method made use of moiré
interferometry. The deformation caused by sectioning and releasing a layer from the
constraints imposed by an adjacent layer was measured. The resulting release of stress
13
could be easily calculated from the deformation strains. The problems resulting from the
cutting method are resolved by obtaining a small strip specimen from the edge of the
larger specimen resulting in a plane stress state.
Another manner in which to use ply sectioning involves the nature of warping seen
in an unbalanced composite laminate. The outside layers of a laminate can be machined
away resulting in the unbalanced, warped structure [3,32]. The resulting warpage is
measured and can be used as an input in classical lamination theory to calculate the
corresponding residual stresses.
2.2.4 First Ply Failure
The first ply failure method makes use of the maximum stress criterion to calculate
the existing residual stresses [16,27]. Hahn and Kim recorded strains and loads while
loading a cross ply, [0/90], specimen to failure. Elastic assumptions were used to
calculate the load at which the first ply failed. This load was compared to the
corresponding strength of a unidirectional specimen and the difference was called the
residual stress. Kam also attempted to consider the visco-elastic effect on this
methodology [25].
2.3 Non-Destructive Techniques
2.3.1 Warpage
Warpage will result in unbalanced, asymmetric multidirectional laminates. The
destructive method of Joh et al. made use of this fact in its calculation of stress from
deformation strain resulting from sectioning. Asymmetric laminates can also be
intentionally manufactured so that the warpage can be measured. This method has been
an area of extensive investigation [8,10,17,22,23,26,27,32,51]. CLT provides the usual
means by which the stresses are calculated from the observed warpage.
14
The combination of warpage and CLT has also been used to devise a method to
measure and quantify polymer matrix cure shrinkage [8,9]. Daniel manufactured a
laminate directly onto an existing and previously cured identical laminate having the
same CTE and manufacturing procedure. Any resulting warpage was strictly the result of
the chemical shrinkage. The curvature was measured and stresses were again calculated
using CLT and the resulting stresses were directly related to chemical shrinkage.
2.3.2 Embedded Strain Gages
Embedded strain gages function by becoming an intimate part of the laminate being
measured. Any deformation of the laminate is directly seen and measured by the strain
gage. The most common strain gages used for this method are the electrical resistance
and fiber optic strain gage.
The first efforts to measure residual stress in this manner were made by Daniel
[5,6,7]. High temperature electrical resistance strain gages were cured directly in the
laminate by placing them between the lamina plies. The strain gages were then directly
exposed to any deformation of the laminate. A compensation method was derived to
account for the varying characteristics of the gage as it went through a given temperature
range. A similar method of compensation is used and described in this thesis. The main
drawback to the method described by Daniel was the introduction of a foreign body to the
laminate. This can drastically alter the material properties at the location as well as create
a void that will lead to incorrect measurements.
Fiber optics represents the second set of strain gages. This gage seems to be
gaining favor in the experimental community. Several recent studies have been
conducted with these gages [2,29,47,49]. In this technique a fiber optic is laid or
embedded in the composite as if it were one of the fibers. Again there is a concern with
15
the introduction of a foreign body into the heart of the laminate. If the optical fiber is not
on the same size scale as the material fiber then there is potential for significant errors
based on the change in the laminate material properties.
2.3.3 X-ray Diffraction
The use of X-ray diffraction in determining stresses was proposed by Predecki and
Barrett in 1979 [40]. Their work was extended to measure residual stresses by Fenn,
Jones, and Wells [11]. With this technique metal particles become part of the matrix
material and experience similar deformations. X-ray measurements are made on the
embedded metal particles and the resulting stresses are correlated to the residual stresses
in the composite. As with the embedding of strain gages, the introduction of metal
particles could result in a change in the laminate properties.
2.3.4 Electrical Resistance
Graphite and Carbon fibers are electrical conductors and therefore a laminate made
from them will have a measure of electrical conductivity and resistance. Researchers are
looking into a technique where the electrical resistance of a laminate can be used to
measure the development of stresses [38,50]. Wang and Chung have observed the
change in resistance across the thickness of a laminate and have set out to correlate the
measured resistance with stresses present in the composite. Park, Lee, and Lee set out to
achieve cure monitoring and strain-stress sensing based on a similar technique.
2.3.5 Cure Referencing Method
The Cure Referencing Method is a technique developed recently by Niu, Ifju, and
Kilday to record process induced strains[20,21,37]. CRM uses the full-field laser based
optical method of moiré interferometry to document strains on the surface of laminates
that initiate during the high temperature curing process. The method involves replicating
16
a high frequency diffraction grating on the specimen while in the autoclave. This method
will be key to the research done in this thesis. A detailed description of the application of
CRM will follow.
CHAPTER 3 CURE REFERENCING METHOD
Several distinct experimental techniques were used in conjunction with one another
to develop residual strain, stress, and coefficient of thermal expansion values across a
wide temperature range. The following chapters will describe in detail the processes and
methods used in the research. These processes and experimental techniques include
material lay-up procedures, basic material property characterization, the Cure
Referencing Method (CRM), and basic strain gage techniques.
3.1 Selected Laminates
The first step in developing a better understanding for the issues at hand was to
manufacture a desired test specimen. Three different composite laminates have been
chosen for study. A composite laminate consists of two or more layers or plies of lamina
arranged in any stacking sequence. Three different stacking sequences of IM7/977-2
were selected for study based on their relative importance. These laminates were:
1. Unidirectional (UNI): [013] 2. Reusable Launch Vehicle (RLV): Quasi-isotropic lay-up [45/903/-45/Ō3]S 3. Optimized Angle Ply (OAP): [(±25)]3S
The unidirectional stacking sequence was selected for its ability to provide insight
into some of the material properties of the material system [19]. It will be seen later how
the strain values obtained from the unidirectional material are used directly in the
analysis to calculate residual strain and stress in the other laminates.
The RLV was, as its name implies, taken directly from the stacking sequence
selected by Lockheed Martin for the design of the X-33 LH2 fuel tanks. It was designed
17
18
as a somewhat quasi-isotropic laminate to allow for maximum strength in all directions.
Quasi-isotropic lay-ups are very common in structural applications of ASCs. However, it
was not designed to account for the loss in strength that is associated with the
development of thermal residual stresses.
The lay-up selected as the optimized angle ply was the result of work done in an
effort to account for thermal stresses while still maintaining a level of mechanical
strength. This effort resulted from a reliability optimization performed by Qu [41], of a
LH2 fuel tank constructed from an IM600/133 graphite/epoxy material system. The
stacking sequence for the tank was [±θ1/±θ2]s. Optimal designs had ply angles θ1 and θ2
near 25o, and nearly equal; the stacking sequence was thus simplified to [±θ]s. Monte
Carlo simulation gave a probability of failure of approximately 1 in 15,000 for the [±25o]s
design. It should be noted that this optimized ply sequence was for a different, but
similar material system.
These laminates were chosen for the study of the residual strain, stress and CTE.
Different sequences were used in the basic material property determination and they will
be described briefly in a following chapter.
3.2 Cure Referencing Method
The CRM is a technique developed at the University of Florida by Dr. Peter Ifju
and Xiaokai Niu. It is a non-destructive novel testing method designed to accurately
determine process induced and residual strains that are associated with the
manufacturing, cure, and thermal loading of an ASC. This method is based on a full-field
measurement technique know as moiré interferometry in which a diffraction grating is
placed on a test piece during the cure cycle. The following section aims to describe CRM
and the procedures behind its implementation.
19
3.3 Moiré Interferometry
Moiré Interferometry is a very precise and sensitive laser based optical technique
that allows the development of a contour map of the in-plane displacements [39]. This
technique provides a displacement sensitivity of 0.417 µm per fringe order. Moiré relies
on the interference properties of light that results in the development of dark and light
bands known as fringes. For this interference to be observed, two diffraction gratings
must be compared to one another. One diffraction grating must be replicated to the
desired part or specimen that is to be deformed while another grating is to remain
unaltered to serve as a reference. This unaltered grating may be one applied to an un-
deformed specimen or the master grating that is used to transfer the pattern to specimens
as described in the next section. The moiré interferometer is tuned with the reference
grating. “Tuned” refers to aligning the interaction of the laser and reference grating in a
manner so that the displacement field is nullified, or no fringes are present. This is
necessary so that only the deformation of the specimen is recorded by the presence of
fringes. The deformation that develops in a specimen can be observed at anytime under
any loading condition as long as the reference grating is still intact. In this study the
deformation given by moiré was only observed for a mechanically unloaded specimen at
room temperature. A four-beam moiré interferometer, schematically represented in
Figure 3-1, was used to record the deformation data.
The interferometer used in this experiment was a 2400 line/mm four-beam system
seen in Figure 3-2. The interferometer consisted of a 10 milliwatt Helium Neon laser
with a 632 nm wavelength and a 108 mm diameter parabolic mirror. The field of view
provided was 38.1mm or 1.5 inches in diameter.
20
Figure 3-1: Schematic of a four-beam moiré interferometer
Figure 3-2: Image of moiré interferometer used in testing
3.4 Diffraction Gratings
Application of a diffraction grating to a part during cure is the central idea and
procedure behind CRM. Before this grating was applied to a specimen it was carefully
21
produced in a several step method derived from the process developed by Niu [37]
described next.
The development of an appropriate diffraction grating to be attached to a specimen
followed these procedures and is seen schematically in Figure 3-3:
5. A grating was replicated at room temperature onto Astrositall glass (ULE) using silicone rubber from an original photoresist master grating. The glass had dimension of 3”x 4.5”x 0.5”. The silicone rubber was a two-part mixture (GE RTV 615). The silicone mixture was centrifuged so that all air bubbles could be removed form the mixture.
6. The silicone grating was then placed into a vacuum deposition chamber where a very thin layer of aluminum was deposited onto its surface. This layer was deposited so that another piece of Astrositall could be easily separated from its surface in a later step.
7. The aluminum coated silicone grating was then used to replicate onto the Astrositall autoclave tool. This grating was made of 3501-6 epoxy. Both pieces of Astrositall were placed into a heating/vacuum chamber where they were heated to 177º C and allowed to soak for approximately one hour. A small piece of 3501-6 was pulled from the freezer and placed into a metal tray where it could be heated. A vacuum was applied to the chamber after heating and liquefying the epoxy for 15 minutes so that any trapped gas bubbles were removed from the epoxy. All three components, the two pieces of Astrositall and the heated epoxy, were removed from the oven. Rapidly, the liquid epoxy was poured into a small pool on the clean Astrositall and then the aluminized silicone grating was placed squarely on top. The two pieces of Astrositall were returned to the oven and a small weight was placed on top so that the grating would be very thin. They were cured at 177º C for 10 hours.
8. The autoclave tool was then separated and prepared for use. The autoclave tool was separated from the silicone tool by use of the instrument seen in Figure 3-4. After separating, the autoclave tool was sent into the vacuum deposition chamber, where two layers of aluminum were cast onto the grating. Each layer of aluminum was separated by a thin layer of dilute Kodak Photo Flo solution that was applied between the first and second deposition of aluminum. The orientation of the grating lines on the tool was determined by the use of an alignment device similar to one described by Post et. Al [39]. A straight line was marked on the tool so that the fibers of the composite specimens were easily aligned with the direction of the grating.
9. A thin layer of 3501-6 was then cast onto the autoclave tool directly on top of the grating surface. A Teflon coated tool, seen in 3.5, was then used to apply the thin film of epoxy. As in a previous step some of the components, the autoclave tool
22
and Teflon coated device, were heated for one hour at 177º C. At several intervals during this hour the Teflon device was removed and the Teflon was tightened to ensure a uniform film of epoxy. After the initial hour, a small piece of 3501-6 was then placed into the metal tray where it was heated for 15 minutes and then vacuumed to remove air bubbles. The components were removed and a thin pool of epoxy was poured onto the grating surface. The Teflon device was lowered into the pool at an angle to push out any possible entrapped air bubbles. The epoxy was cured at 177º C for a period of 10 hours. After cure the autoclave tool was removed from the oven and the Teflon device was disassembled leaving only the sheet of Teflon on the tool. This sheet of Teflon was removed from the surface of the tool by pulling back at a sharp angle.
After completion of the diffraction grating preparation, the grating was transferred
to the specimen by the following method.
3.5 Curing Procedure
As with most advanced composite materials, the laminates in this study were cured
in an autoclave (Figure 3-6). The autoclave is a pressurized oven that has vacuum line
connections inside the pressurized chamber. After selection of an appropriate laminate
the following steps were implemented [18]. It should be noted that the following
procedure was specific to the implementation of CRM. Other laminates were produced
using the same basic procedure and cure cycle with the only difference being that they
are cured without the addition of the ultra-low expansion Astrositall glass and grating
application.
23
Figure 3-3: Replication schematic
24
Figure 3-4: Tool separation device
Figure 3-5: Teflon coated epoxy film casting device
25
10. Lay-up: Individual sheets were cut to size from a roll of prepreg. Those sheets were then arranged and “stacked” in the proper sequence paying heavy attention to maintaining the correct angles.
11. Vacuum bagging: Figure 3-7 shows a schematic of the vacuum bag assembly. The ULE tool with the grating side up was placed at the base of the assembly. The surface was then covered with a non-porous release film. A hole of approximately 1.5” diameter was cut into the release film. The hole allows for the grating to be exposed to the composite surface during the cure cycle. The thickness of the film was important. It was critical that it was as minimal as possible so that it did not indent the surface of the composite during the cure process as the resin began to polymerize. The prepreg laminate was then placed on top of the release film, centered over the ULE tool. The non-porous release film placed between the tool and the laminate prevented the epoxy from curing to the tool. A layer of porous release film was placed on the upper surface of the laminate, which provided two functions. It allowed for excess resin to be drawn off the laminate and into the bleeder cloth, as well as facilitated the separation from the bleeder cloth. The bleeder cloth was a porous material that absorbs excess resin drawn out of the laminate via vacuum. Absorbing this excess resin allowed for an appropriate fiber volume fraction to be achieved. The breather cloth, which was the same material as the bleeder cloth, was used to distribute the vacuum throughout the entire bag. Another layer of non-porous release film was placed between the bleeder and the breather cloths so that excess resin was not transferred into the breather material. This could have resulted in resin being drawn into the vacuum line or clogging the breather cloth which then would have prevented an even vacuum distribution on the laminate during cure. Finally the ULE and laminate assembly were placed into a vacuum bag. The bag was sealed with a sealant tape and placed in the autoclave for curing.
12. Cure preparation: The vacuum bag was placed inside the oven and connected to the vacuum line. Vacuum was applied and a leak check was performed. It was imperative that the vacuum integrity was maintained throughout the necessary portions of the curing process.
13. Cure: Execute the cure cycle appropriate for the given material. The cure cycle for the IM7/977-2 can be seen in 3-8.
26
Figure 3-6: Autoclave oven
Figure 3-7: Vacuum bag assembly schematic
27
3.6 Moiré Measurement Technique
After the specimens have gone through the cure cycle the reflective grating was
present on the surface (Figure 3-9). The specimens were then returned to the moiré
interferometer where the deformation measurements were taken. There are many sources
of potential error in this technique if the proper steps were not taken to insure accurate
measurements.
The first step that must be taken was to properly tune the moiré interferometer.
The virtual grating must be tuned accurately to 2400 lines/mm in both the horizontal and
vertical field, U and V respectively. This was done by directing the second order
diffraction of the reference grating back to the fiber optic tip for all adjustable mirrors.
Then fine-tuning was done for each field by adjusting the U and V field mirrors until both
displayed null fields. The exact position of the master grating was noted so that the
specimen was also placed at the exact same position. Failing to do so would have
introduced errors due to the imperfection in the collimated laser beam.
On each specimen a circle of know radius was scribed into the reflective grating.
This established the gage length over which the fringe numbers would be counted. Each
specimen was then placed into the tuned interferometer with a rotation at 90° from the
reference grating. This rotation was needed because the grating surfaces of the
specimens were the mirror image of the reference grating. The resulting fringes were
photographed using Polaroid ISO 400/27° black and white instant sheet film. Images
were taken and recorded from all of the different laminate specimens at room
temperature.
28
Cure Cycle
-50
0
50
100
150
200
250
300
350
400
0 100 200 300 400 500 600
Minutes
Valu
es
Vac
Temperature (F)
Pressure (psi)
Figure 3-8: IM7/977-2 cure cycle
Figure 3-9: Moiré grating on specimen
29
3.7 Data Analysis
The experimental technique and methods described previously resulted in the
acquisition of the very characteristic pattern of light and dark fringes of moiré
interferometry. These fringe patterns were used to directly determine the in-plane
displacements and strains. The relationship between fringe order N and displacement
was as follows:
fNU x= (3.1)
fN
V y= (3.2)
where f was the frequency of the virtual grating; f=2400 lines/mm.
The absolute surface strain was also determined from the fringe pattern analysis by
the following relations:
1 1xx
NU xNx f x f x
ε ∂ ∆∂ = = = ∂ ∂ ∆
(3.3)
1 1yy
N NV y
x f y f yε
∂ ∆ ∂= = =
∂ ∂ ∆
. (3.4)
These equations were used to determine the strain on the surface of both the
unidirectional and multidirectional laminates. These strains were a result of the thermal
contraction, which was a function of the temperature difference between the cure
temperature and operating temperature; chemical shrinkage of the matrix, which was not
temperature dependent, occurring only during cure; and the thermal expansion of the
autoclave tool, which was neglected in the analysis [19].
The resulting characteristic images of each laminate stacking sequence are
displayed in Appendix A and the corresponding values of the surface strain given by
30
CRM can be seen in Tables 3-1 through 3-3. The values seen in the table represent the
values of surface strain that were present at room temperature. These measurements were
extended throughout the range from cure to LN2 temperatures by a method to be
discussed next.
3.8 Error Sources
Several sources of error were potentially present during the implementation of
CRM. These sources included the improper implementation of moiré interferometry,
manufacturing defects in the laminate specimens, and misalignment of CRM grating
surface. Without taking precautions to avoid possible error sources, the indicated strains
of the CRM method could be significantly misleading.
Table 3-1: Total surface strain on unidirectional laminates at 24°C Specimen x-direction y-direction
UNI-13-06-04 0 -6890 UNI-13-02-03 0 -6500 UNI-13-03-02 0 -7220 UNI-13-B2-02 0 -6955 UNI-13-02-05 0 -6824 Average 0 -6877.8 COV ---- 3.77%
Table 3-2: Total surface strain on optimized angle ply at 24°C
Specimen x-direction y-direction OAP-12-B3-01 787 -4920 OAP-12-03-02 820 -5250 OAP-16-04-01 919 -4950 OAP-16-02-02 1030 -5510 OAP-20-07-05 1033 -5250 Average 917.8 -5176 COV 12.5% 4.72%
31
Table 3-3: Total surface strain on RLV laminate at 24°C Specimen x-direction y-direction
RLV-A1-01 -610 -190 RLV-02-02 -525 -197 RLV-03-03 -525 -220 RLV-05-04 -590 -230 Average -562.5 -209.25 COV 6.78% 7.80%
The implementation of moiré interferometry has many possible sources of error
itself. Improper tuning of the interferometer, misalignment of specimens, improper
specimen orientation and placement, and vibration can all lead to potential problems.
Fortunately, these issues are somewhat easily accounted for and all the proper steps were
taken to avoid improper strain reading as a result of these problems.
Other sources of errors were not as easy to account for. Manufacturing defects are
a common event in laminate production. Inclusions of foreign matter can often lead to
incorrect results. The existence of these inclusions on a large scale is easy to detect while
testing due to the large deviations they can cause from expected or previous results.
Manufacturing the laminates in a clean working area and examining each layer prior to
joining eliminated this problem. One source of error that was not ruled out was
misalignment of the layers themselves during manufacturing. All laminates were cut and
aligned as carefully as possible by hand but the element of human error still existed. It
was possible that misalignments of °± 5.2 through the thickness developed. This
misalignment can result in a change in the characteristics of the laminate.
Similar to the alignment of the fibers during manufacturing, the autoclave tool and
the grating on its surface was aligned by hand onto the specimen. The greatest care was
taken to apply the grating as accurately as possible, but a possible slight offset from the
desired axis of measurement may have resulted. This offset would have lead to the
32
measurement of strain in a direction different than what was desired. For example, if the
strain in the fiber direction after cure is desired and the grating was not aligned perfectly
in the x, or fiber direction, as in the case of the unidirectional laminate, the measured
displacement would be higher than the true displacement.
A very important aspect of accuracy was the ability to read the fringe order
information. From equations (3.3) and (3.4) it is seen that any inaccuracy in the reading
of the correct fringe order would have lead directly to measurement errors. Different
fringe pattern densities often required adjustments on gage factor as the images must be
magnified to see the fringes. Different magnifications assured that all patterns could be
read to less than one fringe order. In the case where the gage length was 0.5”, a misread
of one fringe would have resulted in an error of approximately 33 µε. In the case of a
0.25” gage length a misread of one fringe would have led to double the previous, up to 66
µε.
3.9 Conclusions
The Cure Referencing Method has been applied to find the total surface
displacement on various laminate stacking sequences. The calculated surface strains
represent the sum of the contraction from thermal effects as well as chemical shrinkage
that resulted from polymerization. The strains have been averaged and the coefficient of
variation has been calculated for each stacking sequences. Any COV measurements will
be provided to the optimization group for future reliability analysis.
The experimental results were relatively repeatable. All of the COVs were quite
low with the exception of the x-direction OAP values. This indicated quite consistent
and reliable results.
33
The unidirectional laminates showed the highest value of surface strain, followed
by the OAP, and finally the RLV laminates. The high strain level indicated that the
unidirectional and OAP laminates were relatively free to contract. Conversely, the RLV
laminate showed a very low strain value which indicated a high level of constraint. The
data from these tests are revisited in the combination of CRM and strain gage results in
Chapter 4.
CHAPTER 4 STRAIN GAGE METHODS
The second technique of experimentation that was used in this study was that of
basic thermal expansion testing methods conducted with modern strain gage technology.
This method was used to gather real time surface strain data as the temperature was
varied over the test range. Later it will be shown how information gathered with this
testing procedure was used in conjunction with CRM and material property testing to
develop residual strains, stresses, and CTE as a function of temperature.
4.1 Acquisition Theory
When strain gages are used to acquire strain as a result of temperature change the
output cannot be treated as the true strain on the surface. The output of the gage is
actually termed the thermal output. It is caused by two factors, the change in the
resistivity of the grid alloy with change in temperature and the difference in thermal
expansion coefficients between the gage and the test material [34]. The resulting
resistance change is the sum of resistivity and differential expansion effects are
( )[ TF ]RR
GGSG ∆−+=∆ ααβ (4.1)
where ∆R/R is unit resistance change, βG is the thermal coefficient of resistance of the
grid material, αS - αG is the difference in specimen and grid thermal expansions, FG is the
gage factor of the strain gage, and ∆T is an arbitrary temperature change from a reference
temperature. The strain that is indicated from the change in gage resistance was then
34
35
Ii F
RR∆
=ε (4.2)
where FI is the gage factor setting of the instrument. The gage factor of the instrument
was set to that of the gage so that the resulting thermal output, or strain, output by a gage
on a specimen was
( GSG
GSGT T
Fεε )β
ε −+∆=)/( (4.3)
From this output it was necessary to remove the gage contribution so that only the
surface expansion of the specimen was recorded. This was accomplished by placing the
same gage type from the same lot number onto a reference material of a known
expansion coefficient and expansion curve as a function of temperature. When the
reference material was exposed to identical conditions as the specimen the reference
material gives a thermal output of
( GRG
GRGT T
Fεε )βε −+∆=)/( (4.4)
The gage output was measured on the reference material then the theoretical strain
at a given temperature was removed from the total output yielding the gage contribution
to the thermal output [18]. This output was then subtracted from the thermal output of
the specimen and gage reading, yielding only the thermal induced strain on the surface as
)/()/( RGTSGTS εεε −= (4.5)
The acquired strain readings were later combined with CRM data.
4.2 Specimen Preparation
To ensure the continuity of the measurements, the same specimens used for the
CRM procedure were used for the strain gage experiments. The CRM specimens were
36
sectioned so that the gratings were not damaged, yet a large enough area for strain gage
application was obtained. This yielded specimens of approximately 1” x 4”.
The specimens were conditioned and sanded as recommended by Vishay
Measurements Group for strain gage application. The test covered a wide range of
temperatures that demanded special attention to gage and application techniques. For this
reason M-Bond 610 was selected as the adhesive system and two WK-13-250BG-350
gages were applied to each side, one directed in the x-direction and the other in the
y-direction. These gages were selected for their performance characteristics over the
proposed test temperature range. Each gage was aligned in the proper orientation by use
of mylar tape across the terminals. The gage backing and specimen surface were coated
with M-Bond 610 and were exposed to the air for approximately 15 minutes. After
drying, the gages were pressed down onto the specimen and covered with a thin Teflon
film. Pressure pads were placed over the gages and then they were clamped at 40 psi
with 2” spring clamps. The assembly was then placed in a cool oven and the temperature
was increased to 121º C at a rate of 5º C/min. and then was allowed to cure for 3 ½ hours.
The tape was peeled off of the specimens and they were returned to the oven to be post
cured for two hours at 135º C. A schematic and example of the resulting specimens are
seen in Figures 4-1 and 4-2.
Decoupling the strain gage expansion effects accurately was crucial to maintain the
validity of testing results. The reference materials were prepared following the exact
same procedures, making use of the same gages and bonding techniques, as those used
for the preparation of the composite specimens (Figure 4-3).
37
Figure 4-1: Schematic of strain gage alignment
Figure 4-2: Gauging of sectioned specimen
38
Figure 4-3: Gauging of reference materials
4.3 Testing Procedure
The failure of the X-33 provided motivation for this work as well as the
environmental conditions under which the residual strains, stresses, and CTE were
studied. The composite specimens were observed and investigated over a wide range of
temperatures, from the harsh conditions of the cryogenic realm up to the temperatures at
which they were cured. The following section describes the methods that were used to
test and acquire data in these conditions.
After all preparations were made to the specimens and reference material they were
placed in a Sun Systems Model EC1x environmental chamber. Each test consisted of
three specimens and the reference material being exposed to the desired conditions as
controlled by the environmental chamber. Due to the extensive amount of time that was
necessary to run and collect data, each test was divided into separate cryogenic and
elevated temperature runs.
39
4.3.1 Cryogenic Tests
A LN2 tank was attached to the environmental chamber as seen in Figure 4-4. The
chamber was then able to regulate the temperature by controlling the release of LN2. The
specimens and reference material were then supported on a rack covered with Teflon
(Figure 4-5). The Teflon ensured that the specimens were free to expand and contract
without being exposed to friction that might inhibit this deformation. Lead wires were
passed through a port in the chamber to the data acquisition set up. The chamber door
was returned and sealed while the proper connections were made to the acquisition
hardware.
Each gage was read in a quarter-bridge configuration (Figure 4-6). The lead wires
were passed directly to a STB AD808FB Optim Electronics quarter-bridge completion
module. The completed bridges were directed into a National Instruments SCXI-1322
breakout board attached to a NI SCXI-1122 sixteen-channel Isolated Transducer
Multiplexer Module for Signal Conditioning. The signal conditioning board was part of a
SCXI-1000 four slot chassis that directed the data into a Dell Dimension 8100 through a
NI PCI 6035E data acquisition card. The data and hardware was manipulated by a NI
LabVIEW program.
The chamber temperature was set for 20º C and the specimens were allowed to
soak there for 15 minutes. This temperature was selected as the starting point due to the
material property values of the reference material. The National Institute for Standards
40
Figure 4-4: Environmental chamber and LN2
Figure 4-5: Laminate coupons wired and placed inside environmental chamber
41
Figure 4-6: Quarter-bridge schematic
and Testing has established the thermal expansion data for the reference material, Ti6-
4Al-4V, as a deviation from this temperature. The titanium reference expands as:
( ) 10111.7807.4140.2711.1 362312 ××−+×+×−+− −−−kkk TETETEE (4.6)
This equation is accurate to within 1.5% from temperatures of 24°K to 300°K. A reading
was taken at this point and the corresponding voltages were stored and used as tare
voltages that were removed from all subsequent measurements.
After acquiring the tare voltages at 20º C the temperature in the environmental
chamber was reduced to 5º C. The specimens were again allowed to remain at that
temperature for 15 minutes so that temperature was constant through the thickness.
Another set of readings was taken at this temperature. The LabVIEW program that was
written to manipulate the incoming data took the raw voltages and proceeded to
compensate for several factors. The initial voltages read in the tare measurement were
removed from the new voltage readings. The new gage factor for the corresponding
temperature was calculated and then the raw voltage was converted into strain and
adjusted for transverse sensitivity errors. This was done for all gages inside the chamber.
The converted strain readings represented the total thermal output of the gages and the
gage contribution was decoupled as follows. LabVIEW calculated the actual strain
experienced by the reference material according to equation 4.6 and compared it to what
42
was read. The difference in the readings corresponded to the gage contribution to the
thermal output. This difference was calculated then subtracted from all other strain
readings resulting in the true surface strain on the specimens. This was recorded and
saved versus the temperature.
The same procedure was followed throughout the experiment. Dwell times were
increased as the temperature was lowered since it took more time for the specimens and
reference material to reach thermal equilibrium. Thermal equilibrium was verified by
taking several strain readings within a few minutes of each other. If the drift in the
calculated strain was observed to be very small, on the order of 2 to 5 microstrain per 3
minutes then the final reading was recorded and the experiment proceeded to the next
temperature.
The experiment terminated with the final reading at -190º C. This was the lowest
temperature that was obtainable using this environmental chamber and LN2. The
chamber temperature was returned to room temperature in several steps so as not to
thermally shock the specimens. The specimens and reference material were removed
from the chamber and it was heated to remove all moisture. This procedure was followed
for all specimens.
4.3.2 Elevated Temperature
Following the cryogenic testing the specimens were exposed to elevated
temperatures from room temperature to the upper limit of the composite cure
temperature. The same specimens were used for the elevated temperature tests so that a
curve across the entire test range could be created for each individual specimen. The
only difference was in the solder that was used to connect the lead wires, as the solder
appropriate for exposure to cryogenic temperature is not applicable at elevated
43
temperatures. The conductivity of both solders was very similar, thus the effect of this
change was negligible. The same hardware and software programs were used to collect
the data from elevated testing.
The chamber with specimens and reference material inside was again set at 20º C
where after a dwell time tare readings were taken. This initial point was selected in order
to obtain a continuous graph of the surface strains. It was not set at 20º C to account for
the reference material, as a small piece of Astrositall was gauged and used as a reference
instead of the titanium. The data available for the titanium reference was not valid into
the upper limits of the high temperature test. The Astrositall has an extremely low
expansion coefficient so that in the test range its actual surface strain was treated as zero.
The chamber was heated to 24º C and allowed to dwell there. This temperature
location was necessary since it corresponds to the room temperature at which the CRM
measurements were made. The need for the correspondence in temperature locations will
be discussed later. The dwell time needed for the elevated temperature test was much
longer than that of the cryogenic test due to the slower heat absorption of the Astrositall
reference material.
The LabVIEW program followed the same methodology for the calculation of the
true surface stain. The only difference being that the reference material was different and
had a different expansion curve. The Astrositall was treated as non-expanding
throughout the entire test. Therefore, any indicated strain on the reference material was
immediately subtracted from the specimen readings as the gage thermal contribution.
The temperature intervals began as 15º C increments. After the temperature
reached 90º C the increments were increased to 20º C. The test concluded when the
44
chamber was raised to 182º C, which corresponds to the composite cure temperature. As
before, when conditions began to deviate much from room temperature the dwell times
were increased. Thermal equilibrium was verified with the same technique as previously
discussed.
4.4 Data Analysis and Results
The data obtained directly from the above methods did not directly provide the total
surface strain present the laminate as a function of temperature. What was observed
through the strain gage testing was the relative change in surface strain as the temperature
was varied from 20º C. The strain data has to be coupled with the CRM data in order to
encompass all surface strains.
CRM was able to capture the entire strain induced on the surface of the laminate as
it went through cure and was brought down to room temperature, which was 24º C.
These strain values for the various laminates were highlighted previously (Tables 3-1
through 3-3). The data acquired through the use of strain gages must be referenced to
those values. This was the reason for the strain measurement taken at 24º C by the strain
gages.
The total surface strain as a function of temperature was calculated and plotted in
Microsoft Excel. The total surface strain present at any given temperature was:
24εεεε −+= CRMTlam (4.7)
where єT is the strain calculated by the strain gages at any temperature, єCRM is the strain
recorded by CRM at 24º C for a particular specimen, and є24 is the strain recorded by the
strain gages at 24º C. Figures 4-4 through 4-6 show the resulting total surface strains as a
function of temperature on the various specimens of each laminate sequence.
45
All of the curves demonstrate fairly smooth behavior. The total surface strain results for
different specimens of each stacking sequence are in good agreement with each other and
no large deviations are observed among different specimens of the same laminate and the
average COV across the entire temperature range is acceptable for each (Table 4-1). The
surface strains have been averaged for each laminate (Figure 4-7).
-12000
-10000
-8000
-6000
-4000
-2000
0
2000
-200 -100 0 100 200
Temperature (C)
Mic
rost
rain
UNI-E xUNI-E yUNI-B xUNI-B yUNI-M xUNI-M yUNI-N xUNI-N yUNI-F xUNI-F y
Figure 4-7: Unidirectional laminate surface strain
A crucial observation to be made is that the strains do not return to zero as the
temperature is returned to cure. The cure temperature is often referred to as a stress free
state, hence strain is also expected to be zero at this point. This is only true during the
curing of the laminate because during cure the epoxy undergoes polymerization, which is
a one-time event. The resulting strain from this event, referred to as chemical shrinkage
is always present, is often overlooked, and can contribute a large amount of strain to a
given laminate (Tables 4-2 through 4-4). The stress measurement techniques described
previously, and an analytical prediction tool discussed later, known as Classical
46
Lamination Theory (CLT), do not have the ability to account for this phenomenon. CRM
provides a tool that can accurately predict and calculate this effect.
-12000
-10000
-8000
-6000
-4000
-2000
0
2000
-200 -100 0 100 200
Temperature (C)
Mic
rost
rain
OAP-A x
OAP-A y
OAP-E x
OAP-E y
OAP-H x
OAP-H y
OAP-K x
OAP-K y
OAP-I x
OAP-I y
Figure 4-8: OAP laminate surface strain
-1500-1300-1100
-900-700-500-300-100
100300500
-200 -100 0 100 200
Temperature (C)
Mic
rost
rain
RLV-C x
RLV-C y
RLV-L x
RLV-L y
RLV-G x
RLV-G y
RLV-Q x
RLV-Q y
Figure 4-9: RLV laminate surface strain
The strain data was also transformed into a linear coefficient of thermal expansion
from cure to the testing temperature. The data was taken in 15°C and 20°C intervals
allowing the CTE of the laminates to be calculated as a function of temperature. The
47
change in strain over the temperature interval was divided by that interval from cure to
cryogenic. CTE was calculated for each direction in the various specimens as:
( ))( 12
12
TTCTE xTxT
x −−
=εε
(4.8)
( ))( 12
12
TTCTE yTyT
y −
−=
εε (4.9)
The CTE values as a function of temperature are displayed in Figure 4.11. Over
the test range the change in CTE appears to be quite linear and trend lines were created as
such. The linear fit of the CTE of the unidirectional laminate will later be used to refine
Classical Lamination Theory predictions.
4.5 Error Sources
Just as in the case of CRM any misalignment in the orientations during
manufacturing would have led to incorrect measured values. Also, since the strain gage
readings were combined with those of CRM, any problems that may have been
encountered during the alignment of the displacement gratings would have propagated
into the combined measurements. The most likely source for a large error would have
resulted from the misalignment of a strain gage on the specimen.
Strain gages are extremely sensitive to misalignment when placed on composite
specimens. In some cases a °± 4 misalignment of a strain gage would have led to errors
as large as ± in an axial gage measurement, while a%17 %65± error could exist in a
similar transverse measurement. However, in this case the sensitivity to misalignment
was not as dramatic. On the unidirectional laminates the gages were aligned along the 0º
and 90º directions. It was assumed that enough effort was centered on strain gage
alignment that less than offset was induced. This desired alignment dropped °± 2
48
possible misalignment errors approximately to less than 5% in the case of gages off axis
by . In the case of the OAP laminate a °± 2 °± 2 offset leads to an error of about 3%.
The somewhat quasi-isotropic nature of the RLV laminate again helped to minimize the
effects of gage misalignment keeping errors on the order of 5%.
4.6 Conclusions
A method of measuring relative surface strains as they develop as a function of
temperature has been outlined. Proper compensation methods to account for temperature
effects on measurement techniques have been discussed and automated in a LabVIEW
data acquisition program. Measurements of surface strain have been made across a large
temperature range and the CRM and gage values have been combined in a manner to
account for the total surface strain at any given temperature. The surface strains for all
specimens have been plotted as a function of temperature and the averages of the
different laminates have been compared to each other. The chemical shrinkage
component of the total surface strain has been decoupled. Finally, the average CTE of
the different laminates has been computed as a function of temperature.
The different runs of the various specimens were quite consistent. The surface
strain values were reproducible as a function of temperature from run to run. As in the
results from CRM in chapter 3, the unidirectional and OAP laminates continued to
display a higher strain level than the RLV laminate. The same conclusion is reached, that
the increasing offset of ply angles has lead to a lower strain which resulted from high
constraint imposed in the RLV laminate.
The decoupled chemical shrinkage terms were quite large for the transverse
direction of the unidirectional and OAP laminates. These large values indicate that any
49
work that neglects the chemical shrinkage components will neglect a significant strain
event. These values were not seen in the RLV due to the usual constraint mechanisms.
The CTE was seen to tend toward zero as the temperature was decreased. This
event was observed less in the RLV laminate since the CTE was already quite low, again
due to high constraint. The trend toward zero was expected as molecules will tend to
slow in all motion as the temperature is reduced. The CTE observed in the unidirectional
laminates was taken to be the CTE of the material system and is used in CLT calculations
in Chapter 7 as α1 and α2.
Table 4-1: Average COV of surface strains across temperature range Laminate COV x-direction COV y-direction
UNI ----- 4.26% OAP 13.55% 6.10% RLV 9.59% 10.10%
-12000
-10000
-8000
-6000
-4000
-2000
0
2000
-200 -100 0 100 200
Temperature (C)
Mic
rost
rain
UNI xUNI yOAP xOAP yRLV xRLV y
Figure 4-10: Average of laminate surface strains
50
-10
-5
0
5
10
15
20
25
30
35
-200 -150 -100 -50 0 50 100 150 200
Temperature (°C)
CTE
(mic
rost
rain
/°C)
UNI xUNI yOAP xOAP yRLV xRLV y
Figure 4-11: Coefficient of thermal expansion
Table 4-2: Unidirectional chemical shrinkage induced strain Laminate x-direction y-direction
UNI B ----- ----- UNI E 48.5 -2978.5 UNI F 99.3 -2338.5 UNI M 155.9 -2831.1 UNI N 131.1 -2702.0 Average 108.6 -2712.5
Table 4-3: OAP chemical shrinkage induced strain
Laminate x-direction y-direction OAP A ----- -2164.4 OAP E ----- ----- OAP H 298.5 -1359.5 OAP I 550.2 -2153.1 OAP K 582.1 -2430.7 Average 476.9 2026.9
Table 4-4: RLV chemical shrinkage induced strain
Laminate x-direction y-direction RLV C ----- ----- RLV G ----- ----- RLV L -169.3 29.3 RLV Q -83.8 -47.8 Average -126.55 -9.25
CHAPTER 5 MATERIAL PROPERTY TESTING
Basic material tests include those that provide some of the most elementary yet
critical values that can be assigned to a material. The following experiments were
performed to obtain the Young’s Moduli, E1 and E2, Poisson’s ratio, ν12, and the shear
modulus, G12. These properties are necessary for the prediction of the strains and stresses
provided by Classical Lamination Theory (CLT). They must also be used to transform
the experimentally measured strains into stresses.
The American Society for Testing and Materials (ASTM) book of standards was
used as a guideline for the testing procedures implemented. Standard D 3039 was used as
the model for the E1 and E2 tests, while standard D 5379 was used to determine the shear
modulus.
5.1 E1 Testing
ASTM standard D3039 is a test method designed to determine the in-plane tensile
properties of a polymer matrix composite with high modulus fiber reinforcement [44]. A
panel of unidirectional IM7/977-2 was manufactured as discussed in previous sections
without the previsions made for the implementation of CRM. The panel had dimensions
of 10” in the fiber direction by 6” transversely. The panel was seven layers thick to
approach the ASTM recommended thickness of 0.040”.
After the panel was cured, test coupons were cut from it having nominal
dimensions of 10” x 1”. Each coupon was then sanded and conditioned at its midpoint in
accordance with Vishay Measurements Group techniques for strain gage application.
51
52
One CEA-13-125UT-350 tee gage from Vishay was placed on each side of the coupon
with the standard M-bond 200 cyanoacrylate glue. One gage located on each side allows
for cancellation of bending effects. The branches of the tee were placed precisely in the
fiber and transverse directions allowing for the acquisition of information needed to
calculate both the E1 and ν12 properties.
After the coupons were gauged they were then placed in a MTI Phoenix machine
where they could be loaded (Figure 5-1). Each specimen was loaded axially in the fiber
direction with a crosshead displacement rate of 0.01” per minute. Load versus strain data
was acquired nominally at 200-pound intervals with National Instruments data
acquisition hardware. The hardware consisted of an SCXI-1000 four slot chassis, SCXI-
1322 breakout board, and a SCXI-1122 sixteen-channel Isolated Transducer Multiplexer
Module for Signal Conditioning all read into a Dell Dimension 8100 through a NI PCI
6035E data acquisition card. NI LabVIEW software was used to operate and read the
acquisition hardware.
5.2 E2 Testing
ASTM standard 3039 was also used to establish the E2 testing procedures [44]. In
this case a panel of IM7/977-2 was manufactured with dimensions of 6” in the fiber
direction by 7” in the transverse direction. The panel was 14 layers thick to bring the
thickness nominally to 0.080”.
Again the panel was cut into test coupons after the curing process with the test
coupons nominally of dimensions 7” x 1”. Each specimen was sanded and conditioned
as before at its midpoint. One CEA-06-250UN-350 unidirectional gage was applied to
53
each side of the specimen. The gage was placed on the coupon precisely in the matrix
direction, transverse to the fibers.
Figure 5-1: E1 and E2 testing fixture
54
As before, after the specimens were prepared they were subjected to load in the
MTI Phoenix displacement controlled testing machine. The displacement rate was again
0.01” per minute with load versus strain data acquired at approximately 200-pound
intervals. The same NI data acquisition hardware was used for this procedure as in the E1
testing.
5.3 G12 Testing
As with the previous material property tests, the ASTM book of standards was used
as the guideline for establishing test setup and procedures. ASTM D 5379 is designed to
determine the shear properties of high-modulus fiber reinforced composite materials [45].
A 6” x 6” 13 layer panel of unidirectional IM7/977-2 was manufactured and then cured in
the autoclave with the appropriate cure cycle.
After the panel is cured it was sectioned and machined into the proper shape. The
testing coupons were machined into the form of a rectangular flat strips with two v-
shaped notches centered along the length. A diagram of the appropriate geometry is seen
in Figure 5-2. The panel was sent to TMR Engineering for machining of the correct
geometry.
The v-notched specimens were then prepared for testing through the application of
Vishay micro-measurements N2A-06-C032A-500 shear gages. The specimens were
conditioned appropriately and then one gage was applied to each side precisely in the
area spanning the concentric notches.
The gauged coupons were then placed in the MTI machine with a special testing
fixture seen in Figure 5-3. The specimens were inserted in the fixture and then placed
onto the machine in a manner where the notches were along the line-of-action of the
55
testing machine. The two halves were then tensioned at a rate of 0.01” per minute
while the load-shear strain data was collected using the NI hardware.
5.4 Data Analysis and Results
Before each E1, E2, and G12 test was run the dimensions of the all test coupons were
measured. The stress, σ, in each coupon was calculated as a function of cross sectional
area, A, and the load applied by the MTI machine, P:
AP
=σ . (5.1)
The stress was then plotted versus the strain in each coupon. The slopes of the resulting
graphs (Figures 5-4 and 5-5) were the Young’s moduli of each coupon. The various
moduli were very similar in value and were averaged. The results of each test are seen in
Table 5.1. The E1 testing yielded an average value of 150.2 GPa for the modulus of
elasticity in the fiber direction, while the E2 testing yielded an average of 8.04 GPa for
the transverse or matrix modulus.
The shear specimen results are plotted in Figure 5.6. The shear modulus was
calculated in accordance with ASTM Standard D 5379 where a “shear chord” between
the values of 1000 and 3000 µε was calculated as
γτ
∆∆
=chordG (5.2)
The resulting average value for the shear modulus was 4.77 GPa.
The Poisson’s ratio was also calculated from the E1 test by use of the strain
readings from the transverse gages. The strains displayed by the longitudinal and
transverse gages were compared and Poisson’s ratio was calculated as
56
1
212 ε
εν = . (5.3)
All calculated Poisson’s ratios were very similar and average to 0.342.
Figure 5-2: Shear test specimen schematic
Figure 5-3: Shear Testing Fixture
57
0
50000
100000
150000
200000
250000
0 2000 4000 6000 8000 10000
Microstrain
Stre
ss (p
si) Test 1
Test 2Test 3Test 4
Figure 5-4: E1 stress/strain curves
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 1000 2000 3000 4000 5000 6000 7000 8000
Microstrain
Stre
ss (p
si)
Test 1Test 2Test3
Figure 5-5: E2 stress/strain curves
58
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 2000 4000 6000 8000
Microstrain
Stre
ss (P
a) test 1test 2test 3test 4
Figure 5-6: G12 stress/strain curves
Table 5-1: Results of each tension test Run E1 (GPa) E2 (GPa) υ12 G12 (GPa)
1 151.5 7.67 0.345 4.05 2 151.6 8.40 0.343 5.10 3 147.1 8.05 0.343 4.93 4 150.5 ----- 0.338 5.00 Ave 150.2 8.04 0.342 4.77 COV 1.40% 4.54% 0.87% 10.16%
5.5 Micro-mechanical Methods
A detailed micro-mechanical approach is also being used by others to develop
characteristics of the IM7/977-2 material system [4]. The micro-mechanics approach
uses a unit-cell model of the material system. The unit-cell is modeled and analyzed
using finite element analysis of solid elements. Periodic boundary conditions are applied
to the surface of each cell. This micro-mechanical approach has yielded the E1, E2, ν12,
and G12 properties as well. Table 5-2 compares the results of the micromechanical
approach with experimentally determined values.
59
Table 5-2: Comparison of experimental measurements and micro-mechanics values Material Property
Experimental Average
General Dynamics
Rockwell Micro-mechanics
Max Percent Difference
Min Percent Difference
E1 (GPa) 150.2 169.6 148.2 159 12.1 1.34 E2 (GPa) 8.04 ----- 9.31 9.43 15.9 14.6 ν12 .342 ----- ----- .253 29.9 29.9 G12 (GPa)
4.77 4.80 5.17 4.34 9.45 0.63
5.6 Error Sources
Potential sources for error in the determination of material properties included
many of those listed in pervious chapters. These potential sources were manufacturing
defects, especially layer orientation mistakes, and gage misalignment. Again,
misalignment of the strain gages is the most likely to result in the development of
significant errors.
For the most part the potential errors were reduced to below 5% due to the
orientation of the gage with the fibers in the specific tests. The only case of potential
significant error was in the ν12 measurement made by measuring the transverse strain on
the E1 specimen. Any transverse measurement was much more sensitive to
misalignment, however, again the desired angle helped to alleviate possible error.
Transverse measurements were expected to have less than 10% potential error.
5.7 Conclusions
Various material tests have been conducted using ASTM testing standards. The
material properties of Young’ Modulus in both the fiber and transverse direction have
been calculated as well as the resulting Poison’s ratio. Further testing was conducted to
evaluate the shear modulus of the material system.
60
The data was very consistent and possessed little scatter in most of the
measurements. The COV of the various material properties were very low with the
exception of the shear modulus at 10.16%.
Comparisons were made between the experimentally determined material
properties and those calculated using a micro-mechanics based simulation. This
comparison of material properties yielded widely varying results. The E1 results agreed
quite well and the E2 results were not extremely different. However, the results for ν12
did not yield good agreement. There were several potential sources for the dramatic
differences. One significant source of difference could have been the assumptions made
in the micro-mechanic model. This is a FE based model that is very depended on
boundary conditions. Also this model was applied on a single unit cell assuming a square
packing arrangement. These may have been poor assumptions. Just as likely to yield
differing results were the potential error sources listed previously. All of the samples for
each test were sectioned from one plate specific to that test. If any errors had been made
in the lay-up of the plate then those would have been seen in all tested specimens yielding
a precise but not accurate result. More material testing should be conducted on
specimens from more plates to investigate these differences.
CHAPTER 6 PLY LEVEL RESIDUAL STRAINS AND STRESSES
The residual strains and stresses in a composite result from the mismatch in
contraction of the matrix and fibers. This mismatch in contraction is from the
combination of drastically different CTEs and the chemical shrinkage event that is
limited to the polymer matrix. The follow section will analyze the data acquired from the
previous experimental techniques and present the resulting macro-mechanical residual
strains and stresses.
6.1 Residual Strain Calculation
By determining the state that each ply would reach if the bonds between them were
liberated the stresses that held them together can be defined [18]. This liberated strain
will be referred to as the residual strain. Using the assumptions of laminate theory, where
plane sections remain plane, for symmetric lay-ups, the strain on the surface matches that
at any point in the interior, as long as the location is a few ply thicknesses away from the
free edge.
The surface strain in each laminate is first transformed into its fiber and matrix
components of each ply:
[ ] { 12_2_1_
12_
2_
1_
lamlamlamk
ply
ply
ply
T γεεγεε
⋅=
} (6.1)
where εlam is the total surface strain and [T]k is the transformation matrix of each ply:
61
62
[ ]
−−−=
θθθθθθθθθθθθθθ
22
22
22
sincossincossincossincos2cossin
sincos2sincosT . (6.2)
The residual strain in each ply then becomes:
−
=
12_
2_
1_
12_
2_
1_
12_
2_
1_
ply
ply
lply
uni
uni
uni
kres
res
res
γεε
γεε
γεε
, (6.3)
where {εuni} is the free expansion surface strain measured from the unidirectional
specimens. This surface strain represents the strain that a ply would reach if not
constrained by adjacent layers. These calculations are made for each temperature in the
experimental test range. Matrix properties are significantly weaker than those of the fiber
and the first sign of damage is typically mirocracks in the matrix. For this reason, the
residual strains reported will be those in the transverse direction. The residual strains are
plotted as a function of temperature for each layer in the OAP and RLV laminates
(Figure. 6-1).
6.2 Residual Stress
After residual strain was calculated the stress-strain relations were used to define
the corresponding residual stresses:
kres
res
res
kkres
res
res
QQQQQ
=
12_
2_
1_
66
2221
1211
12_
2_
1_
0000
γεε
τσσ
. (6.4)
The resulting residual stresses in the ply of the OAP and RLV laminates are plotted
versus temperature (Figure 6.2).
63
-1.00E-03
1.00E-03
3.00E-03
5.00E-03
7.00E-03
9.00E-03
1.10E-02
-200 -150 -100 -50 0 50 100 150 200
Temperature (ºC)
Stra
in
OAPRLV 45RLV 90RLV 0
Figure 6-1: Residual strain on selected plies
0.00E+001.00E+072.00E+073.00E+074.00E+075.00E+076.00E+077.00E+078.00E+079.00E+07
-200 -100 0 100 200
Temperature (ºC)
Stre
ss (P
a)
OAPRLV 45RLV 90RLV 0
Figure 6-2: Residual stress level on selected plies
64
6.3 Conclusions
A method to quantify residual strains and stresses based on surface strains has been
discussed. This method has been used to display the resulting residual stresses that were
present in the different laminates. As expected, the high level of constraint that has been
discussed throughout this thesis has lead to high level of residual stress. The OAP has
performed much better than the RLV lay-up in reference to the development of thermally
induced stresses. The often neglected chemical shrinkage has induced as much as 15.4
MPa of residual stress in the RLV laminate.
From these results it has been made apparent that the chemical shrinkage is not a
negligible term and must be determined to fully quantify the extent of residual stress
development. It has also been made quite apparent that there is a great need for the
development of optimization techniques to account for the interaction of both
mechanically and thermally induced stresses.
CHAPTER 7 CLASSICAL LAMINATION THEORY
Classical Lamination Theory (CLT) is a predictive tool that makes it possible to
analyze complex coupling effects that may occur in composites. CLT is able to predict
strains, displacements, and curvatures that develop in a laminate as it is mechanically
loaded as well as those resulting from hygrothermal stresses. It is the thermal
contribution to strain that will be investigated in the following section.
CLT is derived from many assumptions, the most basic of which are that individual
lamina are perfectly bonded together so that they behave as a “unitary, nonhomogeneous,
anisotropic plate” [13]. The most important assumption in CLT is that each ply is
assumed to be in a state of plane stress and that interlaminar stresses are neglected.
7.1 Theory Development
Using the coordinate system in Figure 7-1, the laminated plate geometry and ply
numbering scheme defined in Figure 7-2, along with several basic assumptions, a set of
equations are derived to predict deformations and stresses. The basic assumptions
needed in the derivation are:
14. The plate consists of orthotropic lamina bonded together, with principal material axis directed in arbitrary directions with respect to the x and y axes.
15. Edge lengths of the plate must be much greater than the individual ply thicknesses.
16. The u,v,w displacements are small in comparison to the total plate thickness.
17. The in-plane strains εx, εy, and γxy are much smaller than one.
18. Transverse shear strains γxz and γyz are neglected.
19. The u and v displacements are linear functions of the z coordinate.
65
66
20. The transverse normal strain εz is neglected.
21. Hooke’s Law applies to each ply.
22. Total plate thickness is constant.
23. At the plate surface the transverse shear stresses τxz and τyz are zero.
Assumptions 6 and 7 define the displacements as
),(),(
),(),(
),(),(
02
01
0
yxwyxww
yxzFyxvv
yxzFyxuu
==
+=
+=
(7.1)
where u and are the displacements in the x and y directions at the middle of the plate.
The displacement is the same at any thickness in the plate with the same x and y
coordinates due to assumption 7. Substituting equations 7.1 into the strain-displacement
equations for the transverse shear strains and imposing assumption 5 yields
0 0v
w
0),(
0),(
2
1
=∂∂
+=∂∂
+∂∂
=
=∂∂
+=∂∂
+∂∂
=
ywyxF
yw
zv
xwyxF
xw
zu
yz
xy
γ
γ (7.2)
and
ywyxF
xwyxF
∂∂
−=
∂∂
−=
),(
),(
2
1
(7.3)
67
xy
z
Figure 7-1: Laminate coordinate system
k
321
N
z2
z1 z0
z3
zN-1 zk
zk-1
Middle Surfacet
zN
Figure 7-2: Layer schematic and nomenclature
Combining the strain-displacement relations for in-plane strains with Eqs. (7.1) and (7.3)
results in
xyxyxy
yyy
xxx
zxv
yu
zyv
zxu
κγγ
κεε
κεε
+=∂∂
+∂∂
=
+=∂∂
=
+=∂∂
=
0
0
0
(7.4)
68
where mid-laminate strains are
xv
yuyvx
u
xy
y
x
∂∂
+∂∂
=
∂∂
=
∂∂
=
000
00
00
γ
ε
ε
(7.5)
and mid-laminate curvatures are given by
yxw
yw
xw
xy
y
x
∂∂∂
−=
∂∂
−=
∂∂
−=
2
2
2
2
2
2κ
κ
κ
(7.6)
These represent the curvatures associated with the bending at the middle of the laminate
in the xz and yz planes as well as the out of plane twisting of the middle surface of the
laminate. In our case, as with any symmetric laminate, these curvatures will be zero.
They are left in the derivation to give the solution for any arbitrary stacking sequence
where curvature may be present.
Equation (7.4) provides a solution for the strain at any given distance from mid-
laminate. If the various lamina are of known thickness then the distance of each from
mid-laminate is known and the stress-strain relations for the kth lamina can be expressed
as
+++
=
=
xyxy
yy
xx
kxy
y
x
zzz
QQQQQQQQQ
κγκεκε
τσσ
0
0
0
662616
262212
161211
(7.7)
where
69
[ ] [ ] [ ][ ]TQTQ 1−= (7.8)
and
1266
2112
222
212112
21212
2112
111
1
1
1
GQ
EQ
QEQ
EQ
=−
=
=−
=
−=
νν
ννν
νν
(7.9)
When determining strains and stresses in a laminated plate it is convenient to look
at the forces and moments per unit length as they act on a plate. The force per unit
length, Nx, is given by
( ){ }∑ ∫∫=
− −
==N
k
z
z kx
t
t xxk
k
dzdzN1
2
2 1
σσ (7.10)
and the moment per unit length, , is given by xM
( ){ }∑ ∫∫=
− −
==N
k
z
z kx
t
t xxk
k
zdzdzzM1
2
2 1
σσ (7.11)
where t is the laminate thickness, ( )kxσ is the stress in the kth lamina, zk-1 is the distance
from mid-laminate to inner surface of the kth lamina, and zk is the distance to the outer
surface of the same lamina. Combining (7.7) with both (7.10) and (7.11) yields the
following equations for the force and moment per unit length.
( ) ( ) ( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ) ( ) ( ){ } dzzzQzQzQM
dzzQzQzQN
N
k
z
z xyxykyykxxkx
N
k
z
z xyxykyykxxkx
k
k
k
k
∑∫
∑∫
=
=
−
−
+++++=
+++++=
1
016
012
011
1
016
012
011
1
1
κγκεκε
κγκεκε (7.12)
after rearranging (7.12) the force and moment equations can be written as
70
xyyxxyyxx
xyyxxyyxx
DDDBBBM
BBBAAAN
κκκγεε
κκκγεε
1612110
160
120
11
1612110
160
120
11
+++++=
+++++= (7.13)
where the laminate extensional stiffness is given by
( ) ( ) ( 11
2
2−
=−−== ∑∫ kk
k
N
kij
t
t kijij zzQdzQA ) (7.14)
the laminate coupling stiffness is given by
( ) ( ) ( 21
2
1
2
2 21
−=−
−== ∑∫ kkk
N
kij
t
t kijij zzQdzzQB ) (7.15)
and the laminate bending stiffness is
( ) ( ) ( 31
3
1
2
2
2
31
−=−
−== ∑∫ kkk
N
kij
t
t kijij zzQdzzQD ) (7.16)
Finally the entire set of equations can be expressed as
=
xy
y
x
xy
y
x
xy
y
x
xy
y
x
DDDBBBDDDBBBDDDBBBBBBAAABBBAAABBBAAA
MMMNNN
κκκγεε
0
0
0
662616662616
262211262211
161211161211
662616662616
262212262211
161211161211
(7.17)
The above solution exists for the state in which laminates are not subjected to any
thermal or moisture effects. Together these effects are known as hygrothermal effects. It
is the thermal component of this stress that is of the greatest concern in this case and
therefore the stresses and strains in the laminate must be solved for this hygrothermal
effect.
When the moisture and thermal effects are incorporated the resulting stresses on
each lamina are
71
{ } [ ] { } { } { }( )cTQ kkkkk βαεσ −∆−= (7.18)
and the total lamina strains equal
{ } { } { }κεε zk += 0 (7.19)
Equation (7.19) is substituted into Eqn. (7.18) giving
{ } [ ] { } { } { } { }( )cTzQ kkkk βακεσ −∆−+= 0 (7.20)
The force and moment per unit length are then solved in the same manner but this time
by integrating (7.20).
{ } { } [ ] { } { } { } { }( ){ } [ ]{ } [ ]{ } { } { }{ } { } [ ] { } { } { } { }( ){ } [ ]{ } [ ]{ } { } { }MT
kkkk
MT
kkkk
MMDBM
zdzcTzQzdzM
NNBAN
dzcTzQdzN
−−+=
−∆−+==
−−+=
−∆−+==
∫∫
∫∫
κε
βακεσ
κε
βακεσ
0
0
0
0
(7.21)
The contributions of the thermal and moisture components are
{ } [ ] { } ( ) [ ] { } ( )
{ } [ ] { } [ ] { } ( )11
11
−=
−=
−==
−∆=∆=
∑∫
∑∫
kkkk
N
kkk
M
kkkk
N
kkk
T
zzQccdzQN
zzQTTdzQN
ββ
αα (7.22)
and
{ } [ ] { } ( ) [ ] { } ( )
{ } [ ] { } [ ] { } ( )21
2
1
21
2
1
2
2
−=
−=
−==
−∆
=∆=
∑∫
∑∫
kkkk
N
kkk
M
kkkk
N
kkk
T
zzQcczdzQM
zzQTTzdzQM
ββ
αα (7.23)
Finally, the entire solution can be written in matrix form as
72
=
++++++++++++
xy
y
x
xy
y
x
Mxy
Txyxy
My
Tyy
Mx
Txx
Mxy
Txyxy
My
Tyy
Mx
Txx
DDDBBBDDDBBBDDDBBBBBBAAABBBAAABBBAAA
MMMMMMMMMNNNNNNNNN
κκκγεε
0
0
0
662616662616
262211262211
161211161211
662616662616
262212262211
161211161211
(7.24)
This approach was used to predict the strains on the surface of the laminates and
compare them to those measured via experiment. A MatLAB code was authored,
primarily by Thomas Singer and William Schulz, to encompass all of the assumptions
and calculations involved in CLT (Appendix B). The code was executed at each
temperature in which experimental data was collected. CLT makes its predictions based
on the material properties of individual lamina. The material properties of E1, E2, ν12,
G12, and α1 and α2 were obtained from previous experiments. The CTE values of α1 and
α2 have been calculated as functions of temperature from previous analysis. The apparent
change in CTE is linear and therefore an average CTE across the test range was used as
the input for the CLT program.
7.2 Experimental vs. Analytical Results
CLT results were plotted as a function of temperature and compared to the
experimental results (Figures 7-3through 7-5). It can be seen in all cases that there is
substantial difference between the results of CLT and experimental data.
Several factors could lead to potential differences in the measured and predicted
values. The first potential source is in the assumptions that are made in CLT formulation.
The second and most crucial difference is that CLT lacks the ability to interpret the event
of chemical shrinkage and the effect it has on laminate strain. If the values at the cure
temperature are compared it is seen that CLT predicts no strain at this point, while the
73
combined experimental techniques show strains that are the result of the chemical
shrinkage. Since chemical shrinkage is a one-time event and the resulting strains are an
invariant part of the laminate strain they can be added to the results of CLT. Upon the
introduction of the chemical shrinkage term into the CLT results it was seen that the
graphs collapsed onto each other (Figures 7-6 through 7-8 ).
-12000
-10000
-8000
-6000
-4000
-2000
0
2000
-200 -100 0 100 200
Temperature (ºC)
Mic
rost
rain x CLT
y ClLTx y
Figure 7-3: Unidirectional CLT and experimental results
-10000
-8000
-6000
-4000
-2000
0
2000
-200 -100 0 100 200
Temperature (ºC)
Mic
rost
rain
x CLT
y CLT
x
y
Figure 7-4: OAP CLT and experimental results
74
-1300
-1100
-900
-700
-500
-300
-100
100
-200 -100 0 100 200
Temperature (ºC)
Mic
rost
rain x CLT
y CLTxy
Figure 7-5: RLV CLT and experimental results
The resulting agreement between the CLT and experimental values provides proof
of testing concept and a high degree of measurement accuracy.
Figure 7-6: Unidirectional CLT shifted for chemical shrinkage
75
Figure 7-7: OAP CLT shifted for chemical shrinkage
Figure 7-8: RLV CLT shifted for chemical shrinkage
76
7.3 Conclusions
Classical Laminate theory has been examined and used to predict the surface strains
on the different laminates. This examination has taken place as an attempt to verify the
experimentally determined results. It is a well known fact that the traditional approach to
lamination theory does not have the means to account for chemical shrinkage.
When the two results were compared an interesting phenomenon was observed.
The measured and predicted curves appeared to follow nearly the same trends, separated
by an offset. When the curves were examined it was seen that this offset was almost
exactly equal to that of the chemical shrinkage. When the previously decoupled chemical
shrinkage term was added to the analytical results form CLT the offset in the curves
vanished and the two results were nearly identically. Several substantial conclusions
have been drawn from these results.
First, the fact that the resulting curves coincided with each other throughout the
testing range confirmed that the chemical shrinkage can be treated as a one time event,
completely independent of temperature effects. Secondly, the fact that the two curves
coincided very well provided proof of concept in the strain gage measurements and
temperature compensation technique. Last, and perhaps the most important conclusion
was that the CRM method did indeed provide a means by which the true total strain
present in a laminated composite could be measured. From this result, it is seen that once
the chemical shrinkage term is decoupled and defined, the computer based analytical tool
of CLT can be used to predict the development of strains anywhere in a given
temperature range.
77
APPENDIX A CRM IMAGES OF SPECIMENS
U V
Figure A-1: Unidirectional Laminate Run 1
V, y
U, x
77
78
U V
Figure A-2: Unidirectional Laminate Run 2
V
Figure A-3: Unidirectional Laminate Run 3
V, y
U, x
V, y
U, x
79
V
Figure A-4: Unidirectional Laminate Run 4
V
Figure A-5: Unidirectional Laminate Run 5
V, y
U, x
V, y
U, x
80
U V0
Figure A-6: OAP Laminate Run 1
U V
V, y
U, x
V, y
U, x
Figure A-7: OAP Laminate Run 2
81
U V
Figure A-8: OAP Laminate Run 3
U V
V, y
U, x
V, y
U, x
Figure A-9: OAP Laminate Run 4
82
U V
Figure A-10: OAP Laminate Run 5
U V
V, y
U, x
V, y
U, x
Figure A-11: RLV laminate Run 1
83
U V
Figure A-12: RLV Laminate Run 2
U V
Figure A-13: RLV Laminate Run 3
V, y
U, x
V, y
U, x
84
U V
Figure A-14: RLV Laminate Run 4
V, y
U, x
APPENDIX B MATLAB CLT CODE
Main Routine: laminatestress
function [strain0,curve,stressx,stress1]=laminatestress(plydir,thick,E1,E2,G12,nu12,NM,deltaT,c,alpha12,beta12,plots) % laminatestress uses classical laminate theory to give midplane strain and curvature, and % stress (in both the coordinates of the laminate and of the ply) at the top and bottom of each ply. % All referenced equations are from Gibson, Principles of Composite Material Mechanics, 1994 % strain0: midplane strain, in laminate coords. [e_x;e_y;g_xy]. % curve: curvature. [K_x;K_y;K_xy]. % stressx: stress in laminate coords. stressx is 6 x k, where k is the number of plys. % The kth column gives stress in the x and y directions and shear in the xy plane at % the top and bottom of the kth ply; [s_x_top;s_y_top;s_xy_top;s_x_bot;s_y_bot;s_xy_bot]. % stress1: stress in ply coords. Same as stressx, but the kth column is in the kth ply % coords, with the 1-direction longitudinal, and 2-direction transverse. % plydir: string of the form '[90/+-45/(-+30)3/90bar]s' or column matrix of ply directions. Note % that the variables make more sense with the z-axis down; following this convention, the % first ply is the most negative. % plythick: column vector of ply thicknesses. If thick is a scalar, the thickness is applied to % all plys. % E1: column vector of Young's modulii in the fiber direction of each ply. If E1 is a scalar, % the modulus is applied to all plys. % E2: same as E1, in transverse directions. % G12: same as E1, but shear modulii. % nu12: same as E1, but Poisson's ratios. % NM: column vector of mechanical loading conditions [N_x;N_y,N_xy;M_x;M_y;M_xy] per unit % length. % deltaT: change in temperature from cure temp. % c: moisture concentration. % alpha12: matrix of coeffs of thermal expansion, in material coords. alpha12 % is 2xn or 3xn, where n is the number of plies. If alpha12 is 2x1 or
85
86
% 3x1, the CTEs are assumed to apply to all plies. Row 1 is CTE in the % fiber direction, row 2 is CTE in the transverse direction, and row 3 % is 0. % beta12: matrix of coeffs of hygroscopic expansion. Same format as alpha12. % plots: 1 plots stressx and stress1 through thickness; 0 or blank doesn't plot. % Note: the last 5 inputs are optional. If only one is entered, it is taken to be plots. % If 2 are entered, they are taken as deltaT and alpha. If 3 are entered, they are taken % as deltaT, alpha12, and plots. If 4 are entered, they are taken as deltaT, c, alpha12, % and beta12. %----------Housekeeping---------- if exist('deltaT')==0 deltaT=0; c=0; alpha12=[0;0;0]; beta12=[0;0;0]; plots=0; elseif exist('c')==0 plots=deltaT; deltaT=0; c=0; alpha12=[0;0;0]; beta12=[0;0;0]; elseif exist('alpha12')==0 alpha12=c; c=0; beta12=[0;0;0]; plots=0; elseif exist('beta12')==0 plots=alpha12; alpha12=c; c=0; beta12=[0;0;0]; elseif exist('plots')==0 plots=0; end %Sets variables not assigned in input [a,b]=size(NM); for i=a:5 NM(i+1,1)=0; end %Fills in missing force and moment terms if ischar(plydir) plydir=directions(plydir); end %Converts shorthand ply orientation to matrix
87
%----------Computation------------- [m,n]=size(plydir); [stiff,Q,S,Qbar,Sbar,T,z,z0]=ABD(plydir,thick,E1,E2,G12,nu12); %Calculates ABD matrix [Nh,Mh,alpha,beta]=hygrothermal(deltaT,c,alpha12,beta12,Qbar,T,z0,z); NM=NM+[Nh;Mh]; %Adds hygrothermal forces and moments. Eqs 7.95, 7.96 invstiff=inv(stiff); ek=invstiff*NM; %Solves for midplane strain and curvature. Eq 7.46 strain0=ek(1:3); curve=ek(4:6); %Extracting midplane strain and curvature strainxtop=strain0+z0*curve; %Pesky 0-index problem...see below for k=1:m strainx(:,k)=strain0+z(k)*curve; botstressx(:,k)=Qbar(:,:,k)*(strainx(:,k)-alpha(:,k)*deltaT-beta(:,k)*c); if k==1 topstressx(:,k)=Qbar(:,:,k)*(strainxtop-alpha(:,k)*deltaT-beta(:,k)*c); else topstressx(:,k)=Qbar(:,:,k)*(strainx(:,k-1)-alpha(:,k)*deltaT-beta(:,k)*c); end %Stress in general (x-y) coords. Eq 7.85 botstress1(:,k)=T(:,:,k)*botstressx(:,k); topstress1(:,k)=T(:,:,k)*topstressx(:,k); %Stress in ply (1-2) coords. Eq 2.31 end stressx=[topstressx;botstressx]; stress1=[topstress1;botstress1]; %---------Plotting--------- if plots==1 for k=1:m stressxx(2*k-1,1)=stressx(1,k); stressxx(2*k,1)=stressx(4,k); stressxy(2*k-1,1)=stressx(2,k); stressxy(2*k,1)=stressx(5,k); stressxxy(2*k-1,1)=stressx(3,k); stressxxy(2*k,1)=stressx(6,k); stress11(2*k-1,1)=stress1(1,k); stress11(2*k,1)=stress1(4,k); stress12(2*k-1,1)=stress1(2,k); stress12(2*k,1)=stress1(5,k);
88
stress112(2*k-1,1)=stress1(3,k); stress112(2*k,1)=stress1(6,k); end zplot=z0; for k=1:m zplot(2*k,1)=z(k); if k~=m zplot(2*k+1,1)=z(k); end end scalex=max(stressxx); scaley=max(stressxy); scalexy=max(stressxxy); scale1=max(stress11); scale2=max(stress12); scale12=max(stress112); scalex=max([scalex,scaley,scalexy])+.1*abs(max([scalex,scaley,scalexy])); scale1=max([scale1,scale2,scale12])+.1*abs(max([scale1,scale2,scale12])); scalelox=min(stressxx); scaleloy=min(stressxy); scaleloxy=min(stressxxy); scalelo1=min(stress11); scalelo2=min(stress12); scalelo12=min(stress112); scalelox=min(0,min([scalelox,scaleloy,scaleloxy])-.1*abs(min([scalelox,scaleloy,scaleloxy]))); scalelo1=min(0,min([scalelo1,scalelo2,scalelo12])-.1*abs(min([scalelo1,scalelo2,scalelo12]))); strainxplot=[strain0(1)+curve(1)*z0;strain0(1)+curve(1)*z(m)]; strainyplot=[strain0(2)+curve(2)*z0;strain0(2)+curve(2)*z(m)]; strainxyplot=[strain0(3)+curve(3)*z0;strain0(3)+curve(3)*z(m)]; strainzplot=[z0;z(m)]; figure; plot(strainxplot,strainzplot) title('x-strains throughout the thickness, (x,y) coords') xlabel('Strain') ylabel('Postion') axis([-10e-3,20e-3,z0,z(m)]) figure; plot(strainyplot,strainzplot) title('y-strains throughout the thickness, (x,y) coords') xlabel('Strain') ylabel('Postion') axis([-10e-3,20e-3,z0,z(m)]) figure; plot(strainxyplot,strainzplot) title('Shear strains throughout the thickness, (x,y) coords') xlabel('Strain')
89
ylabel('Postion') axis([-10e-3,20e-3,z0,z(m)]) figure; plot(stressxx,zplot) title('x-stresses throughout the thickness, (x,y) coords') xlabel('Stress') ylabel('Position') axis([scalelox,scalex,z0,z(m)]) figure; plot(stressxy,zplot) title('y-stresses throughout the thickness, (x,y) coords') xlabel('Stress') ylabel('Position') axis([scalelox,scalex,z0,z(m)]) figure; plot(stressxxy,zplot) title('Shear stresses throughout the thickness, (x,y) coords') xlabel('Stress') ylabel('Position') axis([scalelox,scalex,z0,z(m)]) figure; plot(stress11,zplot) title('1-stresses throughout the thickness, ply (1,2) coords') xlabel('Stress') ylabel('Position') axis([scalelo1,scale1,z0,z(m)]) figure; plot(stress12,zplot) title('2-stresses throughout the thickness, ply (1,2) coords') xlabel('Stress') ylabel('Position') axis([scalelo1,scale1,z0,z(m)]) figure; plot(stress112,zplot) title('Shear stresses throughout the thickness, ply (1,2) coords') xlabel('Stress') ylabel('Position') axis([scalelo1,scale1,z0,z(m)]) end
Subroutine: directions
function dir=directions(str) % DIRECTIONS Converts from shorthand ply angle notation % DIRECTIONS(str) outputs a column vector of ply angles as given by str. % str should be a string of the form '[(25)2/+-45/(-+60/0)3/90bar]s' % In extended notation, this is % [25/25/+45/-45/-60/+60/0/-60/+60/0/-60/+60/0/90/0/+60/-60/0/+60/-60/0/+60/-60/-45/+45/25/25] % Always start with [. Use / to seperate ply angles. Use (plys)n to % indicate a set of plys repeated n times. Use +-theta to indicate % +theta/-theta, and -+theta to indicate -theta/+theta. Use bar after % the last ply to indicate a symmetric laminate in which the midplane
90
% passes through the middle of the ply. End the ply sequence with ]. % Use s at the end to indicate symmetry (must be used with bar); t or % nothing denotes that the full ply sequence has been given. % % It may sound confusing, but it should actually be quite natural if % you are familiar with the shorthand notation. % % See also PARENTHESES str=lower(str); len=length(str); if str(len)=='t' & str(len-4:len-2)=='bar' error('Hey, dummy, is it symmetric or not?') end sym=0; oddsym=0; if str(len)=='s' sym=1; str=str(1:len-2); elseif str(len)=='t' str=str(1:len-2); else str=str(1:len-1); len=len+1; end if str(len-4:len-2)=='bar' sym=1; oddsym=1; str=str(2:len-5); else str=str(2:len-2); end len=length(str); str=strcat(str,'/'); slash=findstr('/',str); strpos=1; plypos=1; slashpos=1; plydir=0; while strpos<len+1 if str(strpos:strpos+1)=='+-' plydir(plypos)=str2num(str(strpos+2:slash(slashpos)-1)); plydir(plypos+1)=-str2num(str(strpos+2:slash(slashpos)-1)); plypos=plypos+2; strpos=slash(slashpos)+1; elseif str(strpos:strpos+1)=='-+' plydir(plypos)=-str2num(str(strpos+2:slash(slashpos)-1)); plydir(plypos+1)=str2num(str(strpos+2:slash(slashpos)-1)); plypos=plypos+2;
91
strpos=slash(slashpos)+1; elseif str(strpos)=='(' closepar=findstr(')',str(strpos:len))+strpos-1; slash2=findstr('/',str(closepar(1):len+1))+closepar(1)-1; digits=slash2(1)-closepar(1)-1; [newplydir,slashnum]=parentheses(str(strpos:closepar(1)+1)); if slashpos==1 plydir=newplydir; else plydir=[plydir,newplydir]; end slashpos=slashpos+slashnum; strpos=slash(slashpos)+1; [y,z]=size(plydir); plypos=z+1; else plydir(plypos)=str2num(str(strpos:slash(slashpos)-1)); plypos=plypos+1; strpos=slash(slashpos)+1; end slashpos=slashpos+1; end if sym==1 if oddsym==1 [y,z]=size(plydir); plydir=[plydir(1:z-1),fliplr(plydir)]; else plydir=[plydir,fliplr(plydir)]; end end dir=plydir.'; Subroutine: parentheses
function [plydir,slashnum]=parentheses(str) % PARENTHESES converts parenthetical shorthand ply angle notation % [plydir,slashnum]=PARENTHESES(str) outputs the ply direction plydir % and the number of slashes slashnum in the string str. % % See also DIRECTIONS len=length(str); str=strcat(str(1:len-2),'/',str(len-1:len)); mult=str2num(str(len+1)); slash=findstr('/',str); [z,slashnum]=size(slash); slashnum=slashnum-1; strpos=2; plypos=1; slashpos=1; while strpos<len
92
if str(strpos:strpos+1)~='+-' & str(strpos:strpos+1)~='-+' plydir(plypos)=str2num(str(strpos:slash(slashpos)-1)); plypos=plypos+1; strpos=slash(slashpos)+1; elseif str(strpos:strpos+1)=='+-' plydir(plypos)=str2num(str(strpos+2:slash(slashpos)-1)); plydir(plypos+1)=-str2num(str(strpos+2:slash(slashpos)-1)); plypos=plypos+2; strpos=slash(slashpos)+1; else plydir(plypos)=-str2num(str(strpos+2:slash(slashpos)-1)); plydir(plypos+1)=str2num(str(strpos+2:slash(slashpos)-1)); plypos=plypos+2; strpos=slash(slashpos)+1; end slashpos=slashpos+1; end plydir1=plydir; for a=2:mult plydir=[plydir,plydir1]; end
Subroutine: ABD
function [K,Q,S,Qbar,Sbar,T,z,z0]=ABD(plydir,plythick,E1,E2,G12,nu12) % ABD calculates the laminate stiffness matrix of the form [A,B;B,D]. % All referenced equations are from Gibson, Principles of Composite Material Mechanics, 1994 % [K,Q,S,Qbar,Sbar,T,z,z0]=ABD(plydir,plythick,E1,E2,G12,nu12) % % Inputs: % plydir: string of the form '[90/+-45/(-+30)3/90bar]s' or column matrix of ply directions. Note % that the variables make more sense with the z-axis down; following this convention, the % first ply is the most negative. % plythick: column vector of ply thicknesses. If thick is a scalar, the thickness is applied to % all plys. % E1: column vector of Young's modulii in the fiber direction of each ply. If E1 is a scalar, % the modulus is applied to all plys. % E2: same as E1, in transverse directions. % G12: same as E1, but shear modulii. % nu12: same as E1, but Poisson's ratios. % % Outputs: % K: laminate stiffness (ABD) matrix % Q: ply stiffness. 3D array with Q(:,:,k) the stiffness matrix for the kth ply. % S: ply compliance. Also 3D array, same convention. % Qbar: transformed ply stiffness. Also 3D array, same convention. % Sbar: transformed ply compliance. Also 3D array, same convention.
93
% T: transformation matrix. Also 3D array, same convention. Q, S, Qbar, Sbar, and T are % throughput for use by other functions. % z: column vector of z postions, kth entry is z postion of bottom of kth ply. % z0: z position of top of first ply (most negative). % % See also DIRECTIONS, PLANESTRESS %---------Housekeeping--------- %Converts shorthand ply orientation to matrix if ischar(plydir) plydir=directions(plydir); end %Checks laminate property sizes and orients in column vector form [m,n]=size(plydir); [o,p]=size(plythick); if p ~= 1 plythick = plythick.'; o = p; end [q,r]=size(E1); if r ~= 1 E1 = E1.'; q = r; end [s,t]=size(E2); if t ~= 1 E2 = E2.'; s = t; end [u,v]=size(G12); if v ~= 1 G12 = G12.'; u = v; end [w,x]=size(nu12); if x ~= 1 nu12 = nu12.'; w = x; end %Expands uniform ply properties if o==1 plythick=ones(m,1)*plythick; elseif o~=m error('Thickness should be entered either as a scalar or as a vector of length n, where n is the number of plys in the laminate') end
94
if q==1 E1=ones(m,1)*E1; elseif q~=m error('E1 should be entered either as a scalar or as a vector of length n, where n is the number of plys in the laminate') end if s==1 E2=ones(m,1)*E2; elseif s~=m error('E2 should be entered either as a scalar or as a vector of length n, where n is the number of plys in the laminate') end if u==1 G12=ones(m,1)*G12; elseif u~=m error('G12 should be entered either as a scalar or as a vector of length n, where n is the number of plys in the laminate') end if w==1 nu12=ones(m,1)*nu12; elseif w~=m error('nu12 should be entered either as a scalar or as a vector of length n, where n is the number of plys in the laminate') end %---------Computation----------- %Sets z values for bottom of each ply. z0 is top of 1st ply. z0=-sum(plythick)/2; z(1)=z0+plythick(1); for k=2:m z(k)=z(k-1)+plythick(k); end %Calcs stiffnesses and compliances. Each is placed into a 3-D matrix, where Q(:,:,k) corresponds to the kth ply. for k=1:m [Q(:,:,k),S(:,:,k),Qbar(:,:,k),Sbar(:,:,k),T(:,:,k)]=planestress(E1(k),E2(k),G12(k),nu12(k),plydir(k)); end %Assembling A, B, and D. Eqs 7.38, 7.39, 7.40 for i=1:3 for j=1:3 A(i,j)=Qbar(i,j,1)*(z(1)-z0); B(i,j)=1/2*Qbar(i,j,1)*(z(1)^2-z0^2); D(i,j)=1/3*Qbar(i,j,1)*(z(1)^3-z0^3); for k=2:m A(i,j)=A(i,j)+Qbar(i,j,k)*(z(k)-z(k-1)); B(i,j)=B(i,j)+1/2*Qbar(i,j,k)*(z(k)^2-z(k-1)^2); D(i,j)=D(i,j)+1/3*Qbar(i,j,k)*(z(k)^3-z(k-1)^3); end
95
end end %Putting it all together. K=[A,B;B,D]; Subroutine: planestress
function [Q,S,Qbar,Sbar,T]=planestress(E1,E2,G12,nu12,theta) % PLANESTRESS calculates the stiffness and compliance matrices of a lamina % All referenced equations are from Gibson, Principles of Composite Material Mechanics, % 1994 except as noted. % % Inputs: % E1: Young's modulus in the fiber direction. % E2: Young's modulus in the transverse direction. % G12: Shear modulus. % nu12: Poisson's ratio. % theta: angle between ply fibers (1-direction) and x-direction, in degrees. % % Outputs: % Q: stiffness matrix in 12 (fiber) coordinates. % S: compliance matrix in 12 (fiber) coordinates. % Qbar: transformed stiffness matrix in xy (general) coordinates. % Sbar: transformed compliance matrix in xy (general) coordinates. % T: transformation matrix. %--------Computation------- %Change theta to radians. theta=theta*pi/180; nu21 = nu12*E2/E1; %Calculating S and Q. Eqs 2.24, 2.4, 2.6 S=[1/E1, -nu12/E1, 0; -nu12/E1, 1/E2, 0; 0, 0, 1/G12]; Q=inv(S); %Q = [E1/(1-nu12*nu21), nu12*E2/(1-nu12*nu21), 0;nu12*E2/(1-nu12*nu21), E2/(1-nu12*nu21),0;0,0,G12]; %Transformation matrices. Eq 2.32, Ifju's class notes, 2/7/02 T=[(cos(theta))^2, (sin(theta))^2, 2*cos(theta)*sin(theta); (sin(theta))^2, (cos(theta))^2, -2*cos(theta)*sin(theta); -cos(theta)*sin(theta), cos(theta)*sin(theta), (cos(theta))^2-(sin(theta))^2]; R=[1,0,0;0,1,0;0,0,2]; %Calculating Qbar and Sbar. Ifju's class notes, 2/7/02 Qbar=inv(T)*Q*R*T*inv(R); Sbar=R*inv(T)*inv(R)*S*T;
96
Subroutine: hygrothermal
function [Nh,Mh,alpha,beta]=hygrothermal(deltaT,c,alpha12,beta12,Qbar,T,z0,z) % hygrothermal outputs the thermal and moisture induced forces and moments % on a laminate. % All referenced equations are from Gibson, Principles of Composite Material Mechanics, 1994 % Nh: hygrothermal force. % Mh: hygrothermal moment. % alpha: matrix of coeff thermal expansions in laminate coords. alpha is 3xn, % where n is the number of plies. The kth column is [alpha_x;alpha_y;alpha_xy] % for the kth ply. % beta: same as alpha, but coeff hygroscopic expansions. % deltaT: change in temperature from cure temp. % c: moisture concentration. % alpha12: matrix of coeffs of thermal expansion, in material coords. alpha12 % is 2xn or 3xn, where n is the number of plies. If alpha12 is 2x1 or % 3x1, the CTEs are assumed to apply to all plies. Row 1 is CTE in the % fiber direction, row 2 is CTE in the transverse direction, and row 3 % is 0. % beta12: matrix of coeffs of hygroscopic expansion. Same format as alpha12. % Qbar: transformed stiffness matrix. % T: transformation matrices, 3x3xn. % z0: z position of top of 1st ply. % z: column matrix of z positions of bottom of plies. %--------Housekeeping-------- [m,n]=size(z); %determining number of plies [o,p]=size(alpha12); if o==2 alpha12(3,:)=0; end if p==1 for k=2:n alpha12(:,k)=alpha12(:,1); end end [o,p]=size(beta12); if o==2 beta12(3,:)=0; end if p==1 for k=2:n
97
beta12(:,k)=beta12(:,1); end end %adding shear component to CTE and CHE matrices for calculations % and expanding uniform ply properties %--------Computation-------- for k=1:n alpha(:,k)=inv(T(:,:,k))*alpha12(:,k); beta(:,k)=inv(T(:,:,k))*beta12(:,k); end alpha(3,:)=alpha(3,:)*2; beta(3,:)=beta(3,:)*2; %calculating laminate direction CTEs and CHEs--note that % alpha12*inv(T)=[alpha_x;alpha_y;alpha_xy/2], and the third row % must be multiplied by 2 to give the shear CTE. Same for beta. % Eq. 5.22 Nt = deltaT*Qbar(:,:,1)*alpha(:,1)*(z(1)-z0); Nm = c*Qbar(:,:,1)*beta(:,1)*(z(1)-z0); Mt = deltaT/2*Qbar(:,:,1)*alpha(:,1)*(z(1)^2-z0^2); Mm = c/2*Qbar(:,:,1)*beta(:,1)*(z(1)^2-z0^2); %0-index workaround again...see below for k=2:n Nt = Nt + deltaT*Qbar(:,:,k)*alpha(:,k)*(z(k)-z(k-1)); Nm = Nm + c*Qbar(:,:,k)*beta(:,k)*(z(k)-z(k-1)); Mt = Mt + deltaT/2*Qbar(:,:,k)*alpha(:,k)*(z(k)^2-z(k-1)^2); Mm = Mm + c/2*Qbar(:,:,k)*beta(:,k)*(z(k)^2-z(k-1)^2); end % calculating thermal and hygroscopic forces and moments. Eqs 7.87-7.91 Nh=Nt+Nm; Mh=Mt+Mm; % summing for hygrothermal forces and moments
LIST OF REFERENCES
1. Bert, C. W., and Thompson, G. L., “A Method for Measuring Planar Residual Stresses in Rectangularly Orthotropic Materials,” Journal of Composite Materials, Vol. 2, No. 2, April, pp. 244-253, 1968.
2. Carman, G. P., and Sendechyj, G. P., “Review of the Mechanics of Embedded Optical Sensors,” Journal Composites Technology & Research, Vol. 17, No. 3, pp. 183-193, 1995.
3. Chapman, T. J., Gillespie, J. W., Pipes, R. B., Manson, J. E., "Prediction of Process-Induced Residual Stresses in Thermoplastic Composites,” Journal Composite Materials, Vol. 24, pp. 616-643, 1990.
4. Choi, S., Sankar, B.V., A Micromechanics Method to Predict the Microcracking of the LH2 Composite Tank at Cryogenic Temperature, Department of Mechanical and Aerospace Engineering, University of Florida, 2003.
5. Daniel, I. M., and Liber, T., “Effect of Laminate Construction on Residual Stresses in Graphite/Polyimide Composites,” Experimental Mechanics, Vol. 1, pp. 21-25, 1977.
6. Daniel, I. M., and Liber, T., “Lamination Residual Stresses in Fiber Composites,” IITRI Report D6073-I, for NASA-Lewis Research Center, NASA CR-134826, 1975.
7. Daniel, I. M., Liber, T., and Chamis, C. C., “Measurement of Residual Strains in Boron/Epoxy and Glass/Epoxy laminates,” Composite Reliability, ASTM STP 580, American Society for Testing and Materials, pp. 340-351, 1975.
8. Daniel, I. M., Wang, T., “Determination of Chemical Cure Shrinkage in Composite laminates,” Journal of Composites Technology & Research, Vol. 12, No. 3, pp. 172-176, 1989.
9. Daniel, I. M., Wang, T. –M., Karalekas, D. and Gotro, J. T., “Determination of Chemical Cure Shrinkage in Woven_Glass/Epoxy Laminates,” Annual Technical Conference-Society of Plastics Engineers, New York, USA, May 1-4, pp. 632-634, 1989.
10. Fakuda, H., Takahashi, K., and Toda, S., “Thermal Deformation of Anti-Symmetric Laminates At Cure,” Proceedings of ICCM-10, Whistler, B.C., Canada, August, Vol.3, pp. 141-148, 1995.
98
99
11. Fenn, R. H., Jones, A. M., and Wells, G. M., “X-Ray Diffraction Investigation of Triaxial Residual Stresses in Composite Materials,” Journal Composite Materials, Vol. 27, No. 14, pp. 1338-1351, 1993.
12. Final Report of the X-33 Liquid Hydrogen Tank Test Investigation Team, NASA George C Marshall Space Flight Center, Huntsville, AL 35812, May 2000.
13. Gibson, R. F., Principles of Composite Material Mechanics, McGraw-Hill, New York, New York, 1994.
14. Griffith, A.A., “The Phenomena of Rupture and Flow in Solids,” Philosophical Transactions of the Royal Society, Vol. 221A, pp. 163-198, 1920.
15. Groover, M.P., Fundamentals of Modern Manufacturing, Prentice Hall, Upper Saddle River, New Jersey, 1996.
16. Hahn, H. T., “Residual Stresses in Polymer Matrix Composite Laminates,” Composite Materials, Vol. 10, pp. 265-277, 1976.
17. Hyer, M. W., “Calculation of Room-Temperature Shapes of Unsymmetrical Laminates,” Journal Composite materials, Vol. 15, July, pp. 296-310, 1981.
18. Ifju, P.G., Myers, D., Schulz, W., “Residual Stress and Thermal Expansion of Graphite Epoxy Laminates Subjected to Cryogenic Temperatures,” Proceedings of the American Society for Composites 18th Technical Conference, October 19-23, 2003.
19. Ifju, P.G., Haftka, R., Myers, D., Schulz, W., Singer, T., “Statistics Properties and Reliability Based Optimiztion of Laminates for Cryogenic Environments,” AIAA-2003-1935, 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, 7-10 April 2003.
20. Ifju, P.G., Niu, X., Kilday, B.C., Liu, S.C., Ettinger, S.M., “Residual Strain Measurement in Composites Using the Cure-referencing Method,” Journal of Experimental Mechanics, Vol. 40, No. 1, pp. 22-30, 2000.
21. Ifju, P.G., Niu, X., Kilday, B.C., Liu, S.C., "A Novel Means to Determine Residual Stress In Laminated Composites," Journal of Composite Materials, Vol. 33, No. 16, 1999.
22. Jain, L. K., and Mai, Y. W., “On Residual Stress Induced Distortions during Fabrication of Composite Shells,” Proceedings of the American Society for Composites-Tenth Technical Conference, Santa Monica, CA, USA, Oct. 18-20, pp.261-270, 1995.
23. Jeronimidis, G., and Parkyn, A. T., "Residual Stresses in Carbon Fiber-Thermoplastic Matrix Laminates,” Journal Composites Materials, Vol. 22, pp. 401-415, May 1988.
100
24. Joh, D., Byun, K. Y., and Ha, J., “Thermal Residual Stresses in Thick Graphite/Epoxy Composite Laminates-Uniaxial Approach,” Experimental Mechanics, Vol. 33, No., pp. 70-76, 1993.
25. Kam, T. Y., “Nonlinear and First-Ply Failure Analyses of laminated Composite Cross-Ply Plates,” Journal of Composite Materials, Vol. 29, No.4, pp. 463-482, 1995.
26. Kim, K. S., and Hahn, H.T., “Residual Stress Development during Processing of Graphite/ Epoxy Composite,” Composite Science and Technology, 36, pp. 121-132, 1989.
27. Kim, R. Y., and Hahn, H. T., “Effect of Curing Stresses on the First Ply-failure in Composite Laminates,” Journal Composite Materials, Vol. 13, January, pp. 2-16, 1979.
28. Lake, B. R., Appl, F. J., and Bert, C. W., “An Investigation of the Hole-drilling Technique for Measuring Planar Residual Stress in Rectangularly Orthotropic Materials,” Experimental Mechanics, Vol. 10, June, pp. 233-239, 1970.
29. Lawrence, C., Nelson, D.V., Bennett, T.E., Spingarn, J.R., “Embedded Fiber Optic Sensor Method for Determining Residual Stresses in Fiber-Reinforced Composite Materials,” Journal of Intelligent Material Systems and Structures, Vol. 9, pp. 788-799, 1999.
30. Lee, J., and Czarnek, R., “Measuring Residual Strains in Composites Laminates Using Moiré Interferometry,” Proceedings of the 1991 SEM Conference on Experimental Mechanics, Milwaukee, WI, USA, June 1-3, pp. 405-415, 1991.
31. Lee, J., Czarnek, R., and Guo, Y. “Interferometric Study of Residual Strains in Thick Composites,” Proceedings of the 1989 SEM Conference on Experimental Mechanics, Cambridge, MA, USA, June 7-10, pp. 356-364, 1989.
32. Månson, J. E., and Seferis, J.C. “Process Simulated Laminate (PSL)” A Methodology to Internal Stress Characterization in Advanced Composite Materials,” Journal Composite Materials, Vol. 26, No. 3, pp. 405-431, 1992.
33. Mathar, J., “Determination of Initial Stress by Measuring the Deformations Around Drilled Holes,” Transaction of ASME, Vol. 56, p.249, 1934.
34. “Measurement of Thermal Expansion Coefficient,” Tech Note TN-513, Measurements Group, pp. 1-24.
35. “Measurement of Residual Stresses by the Hole-Drilling Strain-Gage Method,” Tech Note TN-503, Measurement Group, pp. 1-15, 1988.
101
36. Nicoletto, G., “Moiré Interferometry Determination of Residual Stresses in the Presence of Gradients,” Experimental Mechanics, Vol. 31, No. 3, pp. 252-256, 1991.
37. Niu, X., Process Induced Residual Stresses and Dimensional Distortions in Advanced Laminated Composites, University of Florida, 1999.
38. Park, J., Lee, S., Lee, J., “Nondestructive Sensing Evaluation of Single Carbon Fiber/Polymer Composites Using Electro-Micromechanical Technique,” International SAMPE Symposium and Exhibition (Proceedings), Vol. 47, pp190-203, 2002.
39. Post, D., Han, B., and Ifju, P. G., High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials, Springer Verlag, New York, 1994.
40. Predecki, P., and Barrett, C. S., “Stress Measurement in Graphite/Epoxy Composites by X-Ray Diffraction from Fillers,” Journal Composite Materials, Vol. 13, pp. 61-71, 1979.
41. Qu, X., Venkataraman, S., Haftka, R.T., Johnson, T.F., “Reliability, Weight, and Cost Tradeoffs in the Design of Composite Laminates for Cryogenic Environments,” AIAA Paper 2001-1327, Proceedings 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference, Seattle, WA, April, 2001.
42. Rendler, N. J., and Vigness, I., “Hole-drilling Strain-gage Method of Measuring Residual Stresses,” Experimental Mechanics, pp. 577-586, December 1966.
43. Schajer, G. S., and Yang, L., “Residual-stress Measurement in Orthotropic Materials Using the Hole-drilling Method,” Experimental Mechanics, Vol. 34, No. 4, pp. 324-333, 1994.
44. “Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials,” ASTM Standards D3039, Vol. 15.03, pp 98-108.
45. “Standard Test Method for Shear Properties of Composite Materials by the V-Notched Beam Method,” ASTM Standards D5379, Vol. 15.03, pp 226-238.
46. Sunderland, P., Yu, W. J., and Månson, J. E., "A Technique for the Measurement of Process-Induced Internal Stresses in Polymers and Polymer Composites,” Proceedings of ICCM-10, Whistler, B. C., Canada, August, Vol. 3, pp. 125-132, 1995.
47. Tanaka, N., Okabe, Y., Takeda, N., “Temperature-Compensated Strain Measurement Using FBG Sensors Embedded in Composite Laminates,” Proceedings of SPIE-The International Society for Optical Engineering, Vol. 4694, pp. 304-313, 2002.
102
48. “Polymer Matrix Composites: Materials, Usage, Design, and Analysis,” The Composite Materials Handbook Mil 17, Vol. 3, pp. 9.1-9.12, ASTM, West Conshohocken, PA, 2002.
49. Unger, W. J., and Hansen, J. S., “The Effect of Thermal Processing on Residual Strain Development in Unidirectional Graphite Fiber Reinforced PEEK,” Journal Composite Materials, Vol. 27, No. 1, pp. 59-81, 1993.
50. Wang, S., Chung, D.D.L., “Electrical Behavior of Carbon Fiber Polyer-Matrix Composites in the Through-Thickness Direction,” Journal of Materials Science, Vol. 35, pp. 91-100, 2000.
51. Wang, T.-M., Daniel, I. M., and Gotro, J. T., “Thermoviscoelastic Analysis of Residual Stresses and Warpage in Composite Laminates,” Journal of Composite Materials, Vol. 26, No. 6, pp. 883-899, 1992
BIOGRAPHICAL SKETCH
Donald Myers was born on August 26, 1978, in North Miami Beach, Florida. He
lived in Miramar, Florida, an area between Miami and Ft. Lauderdale, where he attended
Miramar High School and graduated as Valedictorian in the spring of 1996. Later that
year he began his undergraduate education at the University of Florida in the Department
of Aerospace Engineering, Mechanics, and Engineering Science. He graduated with
honors in May 2001 with a Bachelor of Science in Aerospace Engineering. After
obtaining the B.S., he continued his education at the University of Florida in August of
2001 pursuing a Master of Science from the Department of Mechanical and Aerospace
Engineering. He served as a Graduate Research Assistant under Dr. Bhavani Sankar in
the Advanced Center for Composites Laboratory, from August 2001 till May of 2002. In
May of 2002 he transitioned to the Experimental Stress Analysis Laboratory where he
has been serving as a Graduate Research Assistant under the tutelage of Dr. Peter Ifju.
While in the Experimental Stress Analysis Lab he has been conducting research into
material characterization of graphite/epoxy composites involved in the storage of
cryogenic fuels.
103